Subgroups — #30865843

Order 17152.

1106 conjugacy classes of subgroups.

The subgroup lattice has 638 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22568842	C₂	1	1	yes	maximal, characteristic, complemented
2	#22562616	C₂	1	1	yes	maximal, characteristic, complemented
3	#22561154	C₂	1	2	yes	maximal, complemented
4	#22561129	C₂	1	2	yes	maximal, complemented
5	#22560658	C₂	1	4	yes	maximal, complemented
6	#22560229	C₂	1	4	yes	maximal, complemented
7	C₆₇×#3444	C₂	1	1	yes	maximal, characteristic
8	C₂×#14524002	C₂²	1	2	yes	—
9	#16863453	—	2	1	no	—
10	#16863453	—	2	1	no	—
11	#16863449	—	2	1	no	—
12	#16863449	—	2	1	no	—
13	#16863420	C₂²	1	2	yes	—
14	C₂×#14523970	C₂²	1	8	yes	—
15	#16863360	C₂²	1	2	yes	—
16	C₄×#11761612	C₂²	1	2	yes	—
17	#16863266	—	2	1	no	—
18	#16863262	—	2	1	no	—
19	#16863256	—	2	1	no	—
20	#16863256	—	2	1	no	—
21	C₄×C₄⋊Dic₆₇	C₂²	1	2	yes	—
22	C₄×C₄⋊Dic₆₇	C₂²	1	2	yes	—
23	#16863155	C₂²	1	8	yes	—
24	#533×C₆₇	C₄	1	1	yes	characteristic, complemented
25	#533×C₆₇	C₄	1	1	yes	characteristic, complemented
26	#533×C₆₇	C₂²	1	1	yes	characteristic
27	#532×C₆₇	C₄	1	1	yes	characteristic, complemented
28	#532×C₆₇	C₄	1	1	yes	characteristic, complemented
29	#532×C₆₇	C₂²	1	1	yes	characteristic
30	#529×C₆₇	—	2	1	no	—
31	#529×C₆₇	—	2	1	no	—
32	#529×C₆₇	—	2	1	no	—
33	#529×C₆₇	—	2	1	no	—
34	#529×C₆₇	C₄	1	2	yes	complemented
35	#529×C₆₇	C₄	1	2	yes	complemented
36	#525×C₆₇	—	2	1	no	—
37	#525×C₆₇	—	2	1	no	—
38	#525×C₆₇	C₂²	1	2	yes	—
39	C₄×#41×C₆₇	—	2	1	no	—
40	C₄×#41×C₆₇	—	2	1	no	—
41	C₄×#41×C₆₇	C₂²	1	2	yes	—
42	C₂²×C₄²×C₆₇	C₂²	1	1	yes	characteristic
43	C₂×#11761612	C₂³	1	4	yes	—
44	#14524002	—	4	4	no	—
45	C₂×C₄⋊Dic₆₇	—	2	2	no	—
46	C₂×C₄⋊Dic₆₇	—	2	2	no	—
47	C₂×C₄⋊Dic₆₇	—	2	2	no	—
48	C₂×C₂²⋊Dic₆₇	—	2	2	no	—
49	C₂×C₄×Dic₆₇	C₂³	1	4	yes	—
50	C₄⋊(C₆₇⋊C₈)	—	2	2	no	—
51	C₄⋊(C₆₇⋊C₈)	—	2	2	no	—
52	#14523970	—	4	8	no	—
53	C₂²⋊(C₆₇⋊C₈)	—	2	2	no	—
54	C₂²⋊(C₆₇⋊C₈)	—	2	2	no	—
55	#14523954	—	2	2	no	—
56	C₄×C₆₇⋊C₈	—	2	2	no	—
57	C₆₇⋊#95	—	2	2	no	—
58	C₆₇⋊#95	—	2	2	no	—
59	C₂×#41×C₆₇	—	2	1	no	—
60	C₂×#41×C₆₇	—	2	1	no	—
61	C₂×#41×C₆₇	C₂³	1	2	yes	—
62	C₂³×C₄×C₆₇	D₄	1	1	yes	characteristic
63	C₂³×C₄×C₆₇	D₄	1	1	yes	characteristic
64	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
65	C₄⋊Q₈×C₆₇	C₂×C₄	1	1	yes	characteristic
66	C₄⋊Q₈×C₆₇	C₂×C₄	1	1	yes	characteristic
67	C₄⋊Q₈×C₆₇	C₂×C₄	1	1	yes	characteristic
68	#127×C₆₇	C₂×C₄	1	1	yes	characteristic
69	#126×C₆₇	—	2	2	no	—
70	#126×C₆₇	—	2	2	no	—
71	#125×C₆₇	—	2	1	no	—
72	#125×C₆₇	—	2	1	no	—
73	#125×C₆₇	C₂×C₄	1	2	yes	—
74	#124×C₆₇	—	2	1	no	—
75	#124×C₆₇	—	2	1	no	—
76	#124×C₆₇	C₂×C₄	1	2	yes	—
77	#123×C₆₇	—	2	2	no	—
78	#123×C₆₇	—	2	2	no	—
79	#123×C₆₇	—	2	2	no	—
80	C₂²⋊Q₈×C₆₇	—	2	1	no	—
81	C₂²⋊Q₈×C₆₇	—	2	1	no	—
82	C₂²⋊Q₈×C₆₇	—	2	1	no	—
83	C₂²⋊Q₈×C₆₇	—	2	1	no	—
84	C₂²⋊Q₈×C₆₇	C₂×C₄	1	2	yes	—
85	C₂²⋊Q₈×C₆₇	C₂×C₄	1	2	yes	—
86	#121×C₆₇	—	2	1	no	—
87	#121×C₆₇	—	2	1	no	—
88	#121×C₆₇	—	2	1	no	—
89	#121×C₆₇	—	2	1	no	—
90	#121×C₆₇	C₂×C₄	1	2	yes	—
91	#121×C₆₇	C₂×C₄	1	2	yes	—
92	#120×C₆₇	—	2	2	no	—
93	C₄×Q₈×C₆₇	—	2	1	no	—
94	C₄×Q₈×C₆₇	—	2	1	no	—
95	C₄×Q₈×C₆₇	—	2	1	no	—
96	C₄×Q₈×C₆₇	—	2	1	no	—
97	C₄×Q₈×C₆₇	C₂×C₄	1	2	yes	—
98	C₄×Q₈×C₆₇	C₂×C₄	1	2	yes	—
99	C₄×D₄×C₆₇	—	2	2	no	—
100	C₄×D₄×C₆₇	—	2	2	no	—
101	C₄×D₄×C₆₇	—	2	2	no	—
102	C₄×D₄×C₆₇	—	2	2	no	—
103	C₄×D₄×C₆₇	—	2	2	no	—
104	C₄×D₄×C₆₇	—	2	2	no	—
105	C₄×D₄×C₆₇	—	2	1	no	—
106	C₄×D₄×C₆₇	—	2	1	no	—
107	C₄×D₄×C₆₇	—	2	1	no	—
108	C₄×D₄×C₆₇	—	2	1	no	—
109	C₄×D₄×C₆₇	C₂×C₄	1	2	yes	—
110	C₄×D₄×C₆₇	C₂×C₄	1	2	yes	—
111	#117×C₆₇	—	2	1	no	—
112	#117×C₆₇	—	2	1	no	—
113	#117×C₆₇	—	2	2	no	—
114	#117×C₆₇	—	2	2	no	—
115	#117×C₆₇	—	2	2	no	—
116	#117×C₆₇	C₂³	1	2	yes	—
117	C₂×C₄²×C₆₇	—	2	2	no	—
118	C₂×C₄²×C₆₇	C₂×C₄	1	1	yes	characteristic
119	C₂×C₄²×C₆₇	C₂×C₄	1	1	yes	characteristic
120	C₂×C₄²×C₆₇	C₂×C₄	1	1	yes	characteristic
121	C₂×C₄²×C₆₇	C₂×C₄	1	1	yes	characteristic
122	C₂×C₄²×C₆₇	D₄	1	1	yes	characteristic
123	C₂×C₄²×C₆₇	D₄	1	1	yes	characteristic
124	C₂×C₄²×C₆₇	D₄	1	1	yes	characteristic
125	C₂×C₄²×C₆₇	D₄	1	1	yes	characteristic
126	C₂×C₄²×C₆₇	D₄	1	1	yes	characteristic
127	C₂×C₄²×C₆₇	D₄	1	1	yes	characteristic
128	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
129	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
130	C₂²×Dic₆₇	—	2	4	no	—
131	#11761612	—	2	4	no	—
132	#11761612	—	2	4	no	—
133	C₂×C₆₇⋊C₈	—	2	4	no	—
134	C₄⋊Dic₆₇	—	4	4	no	—
135	C₄⋊Dic₆₇	—	4	4	no	—
136	C₄⋊Dic₆₇	—	4	4	no	—
137	C₂²⋊Dic₆₇	—	4	4	no	—
138	C₄×Dic₆₇	—	2	8	no	—
139	C₂⁴×C₆₇	C₂×D₄	1	1	yes	characteristic
140	#41×C₆₇	—	2	4	no	—
141	#41×C₆₇	—	2	4	no	—
142	#41×C₆₇	—	4	4	no	—
143	#41×C₆₇	—	4	4	no	—
144	C₂×Q₈×C₆₇	—	2	1	no	—
145	C₂×Q₈×C₆₇	—	2	1	no	—
146	C₂×Q₈×C₆₇	C₂²×C₄	1	2	yes	—
147	C₂×D₄×C₆₇	—	2	2	no	—
148	C₂×D₄×C₆₇	—	2	2	no	—
149	C₂×D₄×C₆₇	—	2	2	no	—
150	C₂×D₄×C₆₇	—	2	1	no	—
151	C₂×D₄×C₆₇	—	2	1	no	—
152	C₂×D₄×C₆₇	C₂²×C₄	1	2	yes	—
153	C₂²×C₄×C₆₇	—	2	2	no	—
154	C₂²×C₄×C₆₇	—	2	2	no	—
155	C₂²×C₄×C₆₇	—	2	2	no	—
156	C₂²×C₄×C₆₇	—	2	2	no	—
157	C₂²×C₄×C₆₇	—	2	2	no	—
158	C₂²×C₄×C₆₇	—	4	2	no	—
159	C₂²×C₄×C₆₇	—	4	2	no	—
160	C₂²×C₄×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
161	C₂²×C₄×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
162	C₂²×C₄×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
163	C₂²×C₄×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
164	C₂²×C₄×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
165	C₂²×C₄×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
166	C₂²×C₄×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
167	C₂²×C₄×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
168	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
169	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
170	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
171	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
172	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
173	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
174	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
175	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
176	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
177	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
178	C₄⋊C₄×C₆₇	—	2	1	no	—
179	C₄⋊C₄×C₆₇	—	2	1	no	—
180	C₄⋊C₄×C₆₇	—	2	1	no	—
181	C₄⋊C₄×C₆₇	—	2	1	no	—
182	C₄⋊C₄×C₆₇	—	2	2	no	—
183	C₄⋊C₄×C₆₇	—	2	2	no	—
184	C₄⋊C₄×C₆₇	—	2	2	no	—
185	C₄⋊C₄×C₆₇	—	2	2	no	—
186	C₄⋊C₄×C₆₇	—	2	2	no	—
187	C₄⋊C₄×C₆₇	—	2	2	no	—
188	C₄⋊C₄×C₆₇	C₂²×C₄	1	2	yes	—
189	C₄⋊C₄×C₆₇	C₂²×C₄	1	2	yes	—
190	C₂²⋊C₄×C₆₇	—	2	2	no	—
191	C₂²⋊C₄×C₆₇	—	2	2	no	—
192	C₂²⋊C₄×C₆₇	—	2	2	no	—
193	C₂²⋊C₄×C₆₇	—	2	2	no	—
194	C₂²⋊C₄×C₆₇	—	2	2	no	—
195	C₂²⋊C₄×C₆₇	—	2	2	no	—
196	C₂²⋊C₄×C₆₇	—	2	2	no	—
197	C₂²⋊C₄×C₆₇	—	2	2	no	—
198	C₂²⋊C₄×C₆₇	—	2	2	no	—
199	C₄²×C₆₇	—	2	2	no	—
200	C₄²×C₆₇	—	2	2	no	—
201	C₄²×C₆₇	—	2	2	no	—
202	C₄²×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
203	C₄²×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
204	C₄²×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
205	C₄²×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
206	C₄²×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
207	C₄²×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
208	C₄²×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
209	C₄²×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
210	C₄²×C₆₇	C₂²×C₄	1	1	yes	characteristic
211	C₄²×C₆₇	C₂²×C₄	1	1	yes	characteristic
212	C₄²×C₆₇	C₂²×C₄	1	1	yes	characteristic
213	C₄²×C₆₇	C₂²×C₄	1	1	yes	characteristic
214	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
215	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
216	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
217	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
218	C₂³×C₆₇	—	2	1	no	—
219	C₂³×C₆₇	—	2	1	no	—
220	C₂³×C₆₇	—	2	2	no	—
221	C₂³×C₆₇	—	4	1	no	—
222	C₂³×C₆₇	—	4	1	no	—
223	C₂³×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
224	C₂³×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
225	C₂³×C₆₇	C₂²×D₄	1	1	yes	characteristic
226	Q₈×C₆₇	—	2	4	no	—
227	Q₈×C₆₇	—	4	2	no	—
228	Q₈×C₆₇	—	4	2	no	—
229	D₄×C₆₇	—	2	4	no	—
230	D₄×C₆₇	—	4	2	no	—
231	D₄×C₆₇	—	4	4	no	—
232	D₄×C₆₇	—	4	4	no	—
233	D₄×C₆₇	—	4	2	no	—
234	D₄×C₆₇	—	4	2	no	—
235	D₄×C₆₇	—	4	2	no	—
236	C₂×C₄×C₆₇	—	2	1	no	—
237	C₂×C₄×C₆₇	—	2	1	no	—
238	C₂×C₄×C₆₇	—	2	2	no	—
239	C₂×C₄×C₆₇	—	2	2	no	—
240	C₂×C₄×C₆₇	—	2	2	no	—
241	C₂×C₄×C₆₇	—	2	2	no	—
242	C₂×C₄×C₆₇	—	2	2	no	—
243	C₂×C₄×C₆₇	—	2	2	no	—
244	C₂×C₄×C₆₇	—	2	2	no	—
245	C₂×C₄×C₆₇	—	2	2	no	—
246	C₂×C₄×C₆₇	—	2	2	no	—
247	C₂×C₄×C₆₇	—	2	2	no	—
248	C₂×C₄×C₆₇	—	2	2	no	—
249	C₂×C₄×C₆₇	—	2	2	no	—
250	C₂×C₄×C₆₇	—	2	2	no	—
251	C₂×C₄×C₆₇	—	4	1	no	—
252	C₂×C₄×C₆₇	—	4	1	no	—
253	C₂×C₄×C₆₇	—	4	1	no	—
254	C₂×C₄×C₆₇	—	4	1	no	—
255	C₂×C₄×C₆₇	—	4	1	no	—
256	C₂×C₄×C₆₇	—	4	1	no	—
257	C₂×C₄×C₆₇	—	4	4	no	—
258	C₂×C₄×C₆₇	—	4	4	no	—
259	C₂×C₄×C₆₇	—	4	4	no	—
260	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
261	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
262	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
263	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
264	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
265	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
266	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
267	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
268	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
269	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
270	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
271	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
272	C₂×C₄×C₆₇	H(5,𝔽₂)	1	2	yes	—
273	C₂×C₄×C₆₇	2₋¹⁺⁴	1	2	yes	—
274	C₂×Dic₆₇	—	4	4	no	—
275	C₂×Dic₆₇	—	4	4	no	—
276	C₂×Dic₆₇	—	4	4	no	—
277	C₆₇⋊C₈	—	4	4	no	—
278	C₂²×C₆₇	—	2	1	no	—
279	C₂²×C₆₇	—	2	1	no	—
280	C₂²×C₆₇	—	2	1	no	—
281	C₂²×C₆₇	—	2	1	no	—
282	C₂²×C₆₇	—	2	1	no	—
283	C₂²×C₆₇	—	2	1	no	—
284	C₂²×C₆₇	—	2	1	no	—
285	C₂²×C₆₇	—	2	1	no	—
286	C₂²×C₆₇	—	4	2	no	—
287	C₂²×C₆₇	—	4	2	no	—
288	C₂²×C₆₇	—	4	2	no	—
289	C₂²×C₆₇	—	4	2	no	—
290	C₂²×C₆₇	—	4	2	no	—
291	C₂²×C₆₇	C₂²×C₂²⋊C₄	1	1	yes	characteristic
292	C₂²×C₆₇	#535	1	1	yes	characteristic
293	C₂²×C₆₇	#536	1	1	yes	characteristic
294	C₄×C₆₇	—	2	1	no	—
295	C₄×C₆₇	—	2	1	no	—
296	C₄×C₆₇	—	2	1	no	—
297	C₄×C₆₇	—	2	1	no	—
298	C₄×C₆₇	—	2	1	no	—
299	C₄×C₆₇	—	2	1	no	—
300	C₄×C₆₇	—	2	1	no	—
301	C₄×C₆₇	—	2	1	no	—
302	C₄×C₆₇	—	4	2	no	—
303	C₄×C₆₇	—	4	2	no	—
304	C₄×C₆₇	—	4	2	no	—
305	C₄×C₆₇	#421	1	2	yes	—
306	C₄×C₆₇	#421	1	2	yes	—
307	C₄×C₆₇	#520	1	2	yes	—
308	C₄×C₆₇	#537	1	2	yes	—
309	Dic₆₇	—	8	4	no	—
310	#21900	—	67	1	no	maximal, Sylow
311	C₂×C₆₇	—	2	1	no	—
312	C₂×C₆₇	—	2	1	no	—
313	C₂×C₆₇	—	4	1	no	—
314	C₂×C₆₇	—	4	1	no	—
315	C₂×C₆₇	—	8	2	no	—
316	C₂×C₆₇	#2311	1	1	yes	characteristic
317	C₂×C₆₇	C₂×#421	1	1	yes	characteristic
318	C₂×C₆₇	C₂×#421	1	1	yes	characteristic
319	#3444	D₆₇	1	1	yes	characteristic
320	#2920	—	67	1	no	—
321	#2285	—	67	1	no	—
322	#1986	—	67	2	no	—
323	#1980	—	67	2	no	—
324	#1897	—	67	4	no	—
325	#1806	—	67	4	no	—
326	C₆₇	#21900	1	1	yes	characteristic, complemented, Sylow
327	#533	Dic₆₇	1	1	yes	characteristic, complemented
328	#533	Dic₆₇	1	1	yes	characteristic, complemented
329	#533	C₂×D₆₇	1	1	yes	characteristic
330	#532	Dic₆₇	1	1	yes	characteristic, complemented
331	#532	Dic₆₇	1	1	yes	characteristic, complemented
332	#532	C₂×D₆₇	1	1	yes	characteristic
333	#529	—	2	1	no	—
334	#529	—	2	1	no	—
335	#529	—	2	1	no	—
336	#529	—	2	1	no	—
337	#529	Dic₆₇	1	2	yes	complemented
338	#529	Dic₆₇	1	2	yes	complemented
339	#525	—	2	1	no	—
340	#525	—	2	1	no	—
341	#525	C₂×D₆₇	1	2	yes	—
342	C₄×#41	—	2	1	no	—
343	C₄×#41	—	2	1	no	—
344	C₄×#41	C₂×D₆₇	1	2	yes	—
345	C₂×#117	—	67	2	no	—
346	C₂²×C₄²	C₂×D₆₇	1	1	yes	characteristic
347	#432	—	134	1	no	—
348	#432	—	134	1	no	—
349	#431	—	134	1	no	—
350	#431	—	134	1	no	—
351	#423	—	67	2	no	—
352	C₂×#104	—	67	8	no	—
353	#407	—	67	2	no	—
354	C₄×M₁₆	—	67	2	no	—
355	#384	—	134	1	no	—
356	#382	—	134	1	no	—
357	#380	—	134	1	no	—
358	#380	—	134	1	no	—
359	C₄×C₄⋊C₄	—	67	2	no	—
360	C₄×C₄⋊C₄	—	67	2	no	—
361	#339	—	67	8	no	—
362	C₂×#41	—	2	1	no	—
363	C₂×#41	—	2	1	no	—
364	C₂×#41	C₂²×D₆₇	1	2	yes	—
365	C₂³×C₄	C₂²⋊D₆₇	1	1	yes	characteristic
366	C₂³×C₄	C₂²⋊D₆₇	1	1	yes	characteristic
367	C₂³×C₄	C₂²×D₆₇	1	1	yes	characteristic
368	C₂×M₁₆	—	67	4	no	—
369	C₄⋊Q₈	C₂×Dic₆₇	1	1	yes	characteristic
370	C₄⋊Q₈	C₂×Dic₆₇	1	1	yes	characteristic
371	C₄⋊Q₈	C₂×Dic₆₇	1	1	yes	characteristic
372	#127	C₂×Dic₆₇	1	1	yes	characteristic
373	#126	—	2	2	no	—
374	#126	—	2	2	no	—
375	#125	—	2	1	no	—
376	#125	—	2	1	no	—
377	#125	C₂×Dic₆₇	1	2	yes	—
378	#124	—	2	1	no	—
379	#124	—	2	1	no	—
380	#124	C₂×Dic₆₇	1	2	yes	—
381	#123	—	2	2	no	—
382	#123	—	2	2	no	—
383	#123	—	2	2	no	—
384	C₂²⋊Q₈	—	2	1	no	—
385	C₂²⋊Q₈	—	2	1	no	—
386	C₂²⋊Q₈	—	2	1	no	—
387	C₂²⋊Q₈	—	2	1	no	—
388	C₂²⋊Q₈	C₂×Dic₆₇	1	2	yes	—
389	C₂²⋊Q₈	C₂×Dic₆₇	1	2	yes	—
390	#121	—	2	1	no	—
391	#121	—	2	1	no	—
392	#121	—	2	1	no	—
393	#121	—	2	1	no	—
394	#121	C₂×Dic₆₇	1	2	yes	—
395	#121	C₂×Dic₆₇	1	2	yes	—
396	#120	—	2	2	no	—
397	C₄×Q₈	—	2	1	no	—
398	C₄×Q₈	—	2	1	no	—
399	C₄×Q₈	—	2	1	no	—
400	C₄×Q₈	—	2	1	no	—
401	C₄×Q₈	C₂×Dic₆₇	1	2	yes	—
402	C₄×Q₈	C₂×Dic₆₇	1	2	yes	—
403	C₄×D₄	—	2	2	no	—
404	C₄×D₄	—	2	2	no	—
405	C₄×D₄	—	2	2	no	—
406	C₄×D₄	—	2	2	no	—
407	C₄×D₄	—	2	2	no	—
408	C₄×D₄	—	2	2	no	—
409	C₄×D₄	—	2	1	no	—
410	C₄×D₄	—	2	1	no	—
411	C₄×D₄	—	2	1	no	—
412	C₄×D₄	—	2	1	no	—
413	C₄×D₄	C₂×Dic₆₇	1	2	yes	—
414	C₄×D₄	C₂×Dic₆₇	1	2	yes	—
415	#117	—	2	1	no	—
416	#117	—	2	1	no	—
417	#117	—	2	2	no	—
418	#117	—	2	2	no	—
419	#117	—	2	2	no	—
420	#117	—	268	4	no	—
421	#117	C₂²×D₆₇	1	2	yes	—
422	C₂×C₄⋊C₄	—	134	2	no	—
423	C₂×C₄⋊C₄	—	134	2	no	—
424	C₂×C₄⋊C₄	—	134	2	no	—
425	C₂×C₂²⋊C₄	—	134	2	no	—
426	C₂×C₄²	—	2	2	no	—
427	C₂×C₄²	—	67	4	no	—
428	C₂×C₄²	C₂×Dic₆₇	1	1	yes	characteristic
429	C₂×C₄²	C₂×Dic₆₇	1	1	yes	characteristic
430	C₂×C₄²	C₂×Dic₆₇	1	1	yes	characteristic
431	C₂×C₄²	C₂×Dic₆₇	1	1	yes	characteristic
432	C₂×C₄²	C₂²⋊D₆₇	1	1	yes	characteristic
433	C₂×C₄²	C₂²⋊D₆₇	1	1	yes	characteristic
434	C₂×C₄²	C₂²⋊D₆₇	1	1	yes	characteristic
435	C₂×C₄²	C₂²⋊D₆₇	1	1	yes	characteristic
436	C₂×C₄²	C₂²⋊D₆₇	1	1	yes	characteristic
437	C₂×C₄²	C₂²⋊D₆₇	1	1	yes	characteristic
438	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
439	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
440	C₄⋊C₈	—	134	2	no	—
441	C₄⋊C₈	—	134	2	no	—
442	#104	—	268	8	no	—
443	C₂²⋊C₈	—	134	2	no	—
444	C₂²⋊C₈	—	134	2	no	—
445	#97	—	134	2	no	—
446	C₄×C₈	—	134	2	no	—
447	#95	—	134	2	no	—
448	#95	—	134	2	no	—
449	C₂⁴	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
450	#41	—	2	4	no	—
451	#41	—	2	4	no	—
452	#41	—	4	4	no	—
453	#41	—	4	4	no	—
454	C₂×Q₈	—	2	1	no	—
455	C₂×Q₈	—	2	1	no	—
456	C₂×Q₈	C₂²×Dic₆₇	1	2	yes	—
457	C₂×D₄	—	2	2	no	—
458	C₂×D₄	—	2	2	no	—
459	C₂×D₄	—	2	2	no	—
460	C₂×D₄	—	2	1	no	—
461	C₂×D₄	—	2	1	no	—
462	C₂×D₄	C₂²×Dic₆₇	1	2	yes	—
463	C₂²×C₄	—	2	2	no	—
464	C₂²×C₄	—	2	2	no	—
465	C₂²×C₄	—	2	2	no	—
466	C₂²×C₄	—	2	2	no	—
467	C₂²×C₄	—	2	2	no	—
468	C₂²×C₄	—	4	2	no	—
469	C₂²×C₄	—	4	2	no	—
470	C₂²×C₄	—	134	4	no	—
471	C₂²×C₄	C₂²⋊Dic₆₇	1	1	yes	characteristic
472	C₂²×C₄	C₂²⋊Dic₆₇	1	1	yes	characteristic
473	C₂²×C₄	C₂²⋊Dic₆₇	1	1	yes	characteristic
474	C₂²×C₄	C₂²⋊Dic₆₇	1	1	yes	characteristic
475	C₂²×C₄	C₂²⋊Dic₆₇	1	1	yes	characteristic
476	C₂²×C₄	C₂²⋊Dic₆₇	1	1	yes	characteristic
477	C₂²×C₄	C₂²⋊Dic₆₇	1	1	yes	characteristic
478	C₂²×C₄	C₂²⋊Dic₆₇	1	1	yes	characteristic
479	C₂²×C₄	C₂²×Dic₆₇	1	1	yes	characteristic
480	C₂²×C₄	C₂²×Dic₆₇	1	1	yes	characteristic
481	C₂²×C₄	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
482	C₂²×C₄	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
483	C₂²×C₄	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
484	C₂²×C₄	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
485	C₂²×C₄	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
486	C₂²×C₄	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
487	C₂²×C₄	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
488	C₂²×C₄	C₂³×D₆₇	1	1	yes	characteristic
489	M₁₆	—	134	4	no	—
490	M₁₆	—	134	4	no	—
491	C₂×C₈	—	134	4	no	—
492	C₄⋊C₄	—	2	1	no	—
493	C₄⋊C₄	—	2	1	no	—
494	C₄⋊C₄	—	2	1	no	—
495	C₄⋊C₄	—	2	1	no	—
496	C₄⋊C₄	—	2	2	no	—
497	C₄⋊C₄	—	2	2	no	—
498	C₄⋊C₄	—	2	2	no	—
499	C₄⋊C₄	—	2	2	no	—
500	C₄⋊C₄	—	2	2	no	—
501	C₄⋊C₄	—	2	2	no	—
502	C₄⋊C₄	—	268	4	no	—
503	C₄⋊C₄	—	268	4	no	—
504	C₄⋊C₄	—	268	4	no	—
505	C₄⋊C₄	C₂²×Dic₆₇	1	2	yes	—
506	C₄⋊C₄	C₂²×Dic₆₇	1	2	yes	—
507	C₂²⋊C₄	—	2	2	no	—
508	C₂²⋊C₄	—	2	2	no	—
509	C₂²⋊C₄	—	2	2	no	—
510	C₂²⋊C₄	—	2	2	no	—
511	C₂²⋊C₄	—	2	2	no	—
512	C₂²⋊C₄	—	2	2	no	—
513	C₂²⋊C₄	—	2	2	no	—
514	C₂²⋊C₄	—	2	2	no	—
515	C₂²⋊C₄	—	2	2	no	—
516	C₂²⋊C₄	—	268	4	no	—
517	C₄²	—	2	2	no	—
518	C₄²	—	2	2	no	—
519	C₄²	—	2	2	no	—
520	C₄²	—	134	8	no	—
521	C₄²	C₂²⋊Dic₆₇	1	1	yes	characteristic
522	C₄²	C₂²⋊Dic₆₇	1	1	yes	characteristic
523	C₄²	C₂²⋊Dic₆₇	1	1	yes	characteristic
524	C₄²	C₂²⋊Dic₆₇	1	1	yes	characteristic
525	C₄²	C₂²⋊Dic₆₇	1	1	yes	characteristic
526	C₄²	C₂²⋊Dic₆₇	1	1	yes	characteristic
527	C₄²	C₂²⋊Dic₆₇	1	1	yes	characteristic
528	C₄²	C₂²⋊Dic₆₇	1	1	yes	characteristic
529	C₄²	C₂²×Dic₆₇	1	1	yes	characteristic
530	C₄²	C₂²×Dic₆₇	1	1	yes	characteristic
531	C₄²	C₂²×Dic₆₇	1	1	yes	characteristic
532	C₄²	C₂²×Dic₆₇	1	1	yes	characteristic
533	C₄²	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
534	C₄²	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
535	C₄²	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
536	C₄²	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
537	C₂³	—	2	1	no	—
538	C₂³	—	2	1	no	—
539	C₂³	—	2	2	no	—
540	C₂³	—	4	1	no	—
541	C₂³	—	4	1	no	—
542	C₂³	C₂×C₂²⋊Dic₆₇	1	1	yes	characteristic
543	C₂³	C₂×C₂²⋊Dic₆₇	1	1	yes	characteristic
544	C₂³	C₂²×C₂²⋊D₆₇	1	1	yes	characteristic
545	Q₈	—	2	4	no	—
546	Q₈	—	4	2	no	—
547	Q₈	—	4	2	no	—
548	D₄	—	2	4	no	—
549	D₄	—	4	2	no	—
550	D₄	—	4	4	no	—
551	D₄	—	4	4	no	—
552	D₄	—	4	2	no	—
553	D₄	—	4	2	no	—
554	D₄	—	4	2	no	—
555	C₂×C₄	—	2	1	no	—
556	C₂×C₄	—	2	1	no	—
557	C₂×C₄	—	2	2	no	—
558	C₂×C₄	—	2	2	no	—
559	C₂×C₄	—	2	2	no	—
560	C₂×C₄	—	2	2	no	—
561	C₂×C₄	—	2	2	no	—
562	C₂×C₄	—	2	2	no	—
563	C₂×C₄	—	2	2	no	—
564	C₂×C₄	—	2	2	no	—
565	C₂×C₄	—	2	2	no	—
566	C₂×C₄	—	2	2	no	—
567	C₂×C₄	—	2	2	no	—
568	C₂×C₄	—	2	2	no	—
569	C₂×C₄	—	2	2	no	—
570	C₂×C₄	—	4	1	no	—
571	C₂×C₄	—	4	1	no	—
572	C₂×C₄	—	4	1	no	—
573	C₂×C₄	—	4	1	no	—
574	C₂×C₄	—	4	1	no	—
575	C₂×C₄	—	4	1	no	—
576	C₂×C₄	—	4	4	no	—
577	C₂×C₄	—	4	4	no	—
578	C₂×C₄	—	4	4	no	—
579	C₂×C₄	—	268	4	no	—
580	C₂×C₄	—	268	4	no	—
581	C₂×C₄	—	268	4	no	—
582	C₂×C₄	C₂×C₂²⋊Dic₆₇	1	1	yes	characteristic
583	C₂×C₄	C₂×C₂²⋊Dic₆₇	1	1	yes	characteristic
584	C₂×C₄	C₂×C₂²⋊Dic₆₇	1	1	yes	characteristic
585	C₂×C₄	C₂×C₂²⋊Dic₆₇	1	1	yes	characteristic
586	C₂×C₄	C₂×C₂²⋊Dic₆₇	1	1	yes	characteristic
587	C₂×C₄	C₂×C₂²⋊Dic₆₇	1	1	yes	characteristic
588	C₂×C₄	C₂×C₂²⋊Dic₆₇	1	1	yes	characteristic
589	C₂×C₄	C₂×C₂²⋊Dic₆₇	1	1	yes	characteristic
590	C₂×C₄	C₂×C₂²⋊Dic₆₇	1	1	yes	characteristic
591	C₂×C₄	C₂×C₂²⋊Dic₆₇	1	1	yes	characteristic
592	C₂×C₄	C₂³×Dic₆₇	1	1	yes	characteristic
593	C₂×C₄	C₂²×C₂²⋊D₆₇	1	1	yes	characteristic
594	C₂×C₄	#14524089	1	2	yes	—
595	C₂×C₄	#14524090	1	2	yes	—
596	C₈	—	268	4	no	—
597	C₂²	—	2	1	no	—
598	C₂²	—	2	1	no	—
599	C₂²	—	2	1	no	—
600	C₂²	—	2	1	no	—
601	C₂²	—	2	1	no	—
602	C₂²	—	2	1	no	—
603	C₂²	—	2	1	no	—
604	C₂²	—	2	1	no	—
605	C₂²	—	4	2	no	—
606	C₂²	—	4	2	no	—
607	C₂²	—	4	2	no	—
608	C₂²	—	4	2	no	—
609	C₂²	—	4	2	no	—
610	C₂²	C₂²×C₂²⋊Dic₆₇	1	1	yes	characteristic
611	C₂²	#16863960	1	1	yes	characteristic
612	C₂²	#16863967	1	1	yes	characteristic
613	C₄	—	2	1	no	—
614	C₄	—	2	1	no	—
615	C₄	—	2	1	no	—
616	C₄	—	2	1	no	—
617	C₄	—	2	1	no	—
618	C₄	—	2	1	no	—
619	C₄	—	2	1	no	—
620	C₄	—	2	1	no	—
621	C₄	—	4	2	no	—
622	C₄	—	4	2	no	—
623	C₄	—	4	2	no	—
624	C₄	—	536	4	no	—
625	C₄	#16863413	1	2	yes	—
626	C₄	#16863413	1	2	yes	—
627	C₄	#16863879	1	2	yes	—
628	C₄	#16863974	1	2	yes	—
629	C₂	—	2	1	no	—
630	C₂	—	2	1	no	—
631	C₂	—	4	1	no	—
632	C₂	—	4	1	no	—
633	C₂	—	8	2	no	—
634	C₂	#22562801	1	1	yes	characteristic
635	C₂	C₂×#16863413	1	1	yes	characteristic
636	C₂	C₂×#16863413	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂²×C₂²⋊D₆₇
commutator	C₂×C₄×C₆₇	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₆₇
Fitting, prime radical	C₆₇×#3444	C₂
socle	C₂²×C₆₇	C₂²×C₂²⋊C₄
cosocle	C₂²×C₄×C₆₇	C₂⁴
hypercenter	#3444	D₆₇
hypocenter	C₆₇	#21900
LCS	G ▷ C₂×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ #3444

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
1. #22568842	C₂	semidirect
2. #22562616	C₂	semidirect
3. #22561154	C₂	semidirect
4. #22561129	C₂	semidirect
5. #22560658	C₂	semidirect
6. #22560229	C₂	semidirect
24. #533×C₆₇	C₄	semidirect
25. #533×C₆₇	C₄	semidirect
27. #532×C₆₇	C₄	semidirect
28. #532×C₆₇	C₄	semidirect
34. #529×C₆₇	C₄	semidirect
35. #529×C₆₇	C₄	semidirect
326. C₆₇	#21900	semidirect
327. #533	Dic₆₇	semidirect
328. #533	Dic₆₇	semidirect
330. #532	Dic₆₇	semidirect
331. #532	Dic₆₇	semidirect
337. #529	Dic₆₇	semidirect
338. #529	Dic₆₇	semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
#22568842	2
#22562616	2
#22561154	2
#22561129	2
#22560658	2
#22560229	2
C₆₇×#3444	2
#21900	67