Subgroups — #30825363

Order 17152.

688 conjugacy classes of subgroups.

The subgroup lattice has 500 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22569176	C₂	1	1	yes	maximal, characteristic, complemented
2	#41×C₆₇⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
3	#22559060	C₂	1	2	yes	maximal, complemented
4	#22559053	C₂	1	2	yes	maximal, complemented
5	#22559046	C₂	1	2	yes	maximal, complemented
6	#22559039	C₂	1	2	yes	maximal, complemented
7	#22558957	C₂	1	1	yes	maximal, characteristic, complemented
8	#22558591	C₂	1	1	yes	maximal, characteristic, complemented
9	#22558591	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×#16863118	C₂	1	1	yes	maximal, characteristic, complemented
11	C₄×#135×C₆₇	C₂	1	1	yes	maximal, characteristic
12	#16863527	C₂²	1	2	yes	—
13	Q₈×C₆₇⋊C₈	C₂²	1	2	yes	—
14	#16863474	C₂²	1	2	yes	—
15	#16863468	—	2	2	no	—
16	D₄×C₆₇⋊C₈	—	2	2	no	—
17	D₄×C₆₇⋊C₈	C₂²	1	2	yes	—
18	#16863449	—	2	1	no	—
19	#16863449	—	2	1	no	—
20	#16863449	C₂²	1	1	yes	characteristic
21	#16863449	C₂²	1	1	yes	characteristic
22	C₄×#11761612	—	2	1	no	—
23	C₂×C₄×C₆₇⋊C₈	—	2	1	no	—
24	C₂×C₄×C₆₇⋊C₈	C₂²	1	1	yes	characteristic
25	C₂×C₄×C₆₇⋊C₈	C₂²	1	1	yes	characteristic
26	#16863149	C₂²	1	2	yes	—
27	#16863147	C₂²	1	2	yes	—
28	Q₈⋊(C₆₇⋊C₈)	C₂²	1	2	yes	—
29	Q₈⋊(C₆₇⋊C₈)	C₂²	1	2	yes	—
30	D₄⋊(C₆₇⋊C₈)	C₂²	1	2	yes	—
31	D₄⋊(C₆₇⋊C₈)	C₂²	1	2	yes	—
32	#16863118	C₂²	1	2	yes	—
33	#16863118	C₂²	1	2	yes	—
34	C₂×#135×C₆₇	C₄	1	1	yes	characteristic
35	C₄×#41×C₆₇	C₂²	1	1	yes	characteristic
36	C₄×#41×C₆₇	C₂²	1	1	yes	characteristic
37	C₄×Q₁₆×C₆₇	C₄	1	1	yes	characteristic
38	C₄×Q₁₆×C₆₇	C₂²	1	1	yes	characteristic
39	C₄×SD₁₆×C₆₇	C₄	1	1	yes	characteristic
40	C₄×SD₁₆×C₆₇	C₄	1	1	yes	characteristic
41	C₄×SD₁₆×C₆₇	C₂²	1	1	yes	characteristic
42	C₄×SD₁₆×C₆₇	C₂²	1	1	yes	characteristic
43	C₄×D₈×C₆₇	C₄	1	1	yes	characteristic
44	C₄×D₈×C₆₇	C₂²	1	1	yes	characteristic
45	#427×C₆₇	C₄	1	1	yes	characteristic
46	#416×C₆₇	C₄	1	1	yes	characteristic
47	#416×C₆₇	C₄	1	1	yes	characteristic
48	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
49	C₂×#11761612	—	2	2	no	—
50	C₂²×C₆₇⋊C₈	—	2	2	no	—
51	C₂²×C₆₇⋊C₈	—	2	1	no	—
52	C₂²×C₆₇⋊C₈	D₄	1	2	yes	—
53	C₄⋊(C₆₇⋊C₈)	—	2	2	no	—
54	C₄⋊(C₆₇⋊C₈)	—	2	2	no	—
55	C₄⋊(C₆₇⋊C₈)	C₂³	1	2	yes	—
56	C₄⋊(C₆₇⋊C₈)	C₂³	1	2	yes	—
57	C₂²⋊(C₆₇⋊C₈)	—	2	2	no	—
58	C₂²⋊(C₆₇⋊C₈)	—	2	2	no	—
59	C₂²⋊(C₆₇⋊C₈)	—	2	1	no	—
60	C₂²⋊(C₆₇⋊C₈)	—	2	1	no	—
61	#14523954	—	2	2	no	—
62	C₄×C₆₇⋊C₈	—	2	2	no	—
63	C₄×C₆₇⋊C₈	—	2	1	no	—
64	C₄×C₆₇⋊C₈	—	2	1	no	—
65	C₄×C₆₇⋊C₈	—	2	1	no	—
66	C₄×C₆₇⋊C₈	D₄	1	1	yes	characteristic
67	C₄×C₆₇⋊C₈	D₄	1	1	yes	characteristic
68	C₄×C₆₇⋊C₈	C₂³	1	2	yes	—
69	C₄×C₆₇⋊C₈	C₂³	1	2	yes	—
70	C₂×#41×C₆₇	C₂×C₄	1	1	yes	characteristic
71	C₂×#41×C₆₇	C₂×C₄	1	1	yes	characteristic
72	#135×C₆₇	—	2	4	no	—
73	C₂×Q₁₆×C₆₇	C₂×C₄	1	1	yes	characteristic
74	C₂×SD₁₆×C₆₇	C₂×C₄	1	1	yes	characteristic
75	C₂×SD₁₆×C₆₇	C₂×C₄	1	1	yes	characteristic
76	C₂×D₈×C₆₇	C₂×C₄	1	1	yes	characteristic
77	C₂²×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
78	C₂²×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
79	C₄×Q₈×C₆₇	C₂×C₄	1	1	yes	characteristic
80	C₄×Q₈×C₆₇	C₂×C₄	1	1	yes	characteristic
81	C₄×Q₈×C₆₇	C₂³	1	1	yes	characteristic
82	C₄×Q₈×C₆₇	C₂³	1	1	yes	characteristic
83	C₄×D₄×C₆₇	—	2	1	no	—
84	C₄×D₄×C₆₇	—	2	1	no	—
85	C₄×D₄×C₆₇	—	2	1	no	—
86	C₄×D₄×C₆₇	—	2	1	no	—
87	C₄×D₄×C₆₇	C₂×C₄	1	1	yes	characteristic
88	C₄×D₄×C₆₇	C₂×C₄	1	1	yes	characteristic
89	C₄×D₄×C₆₇	C₂³	1	1	yes	characteristic
90	C₄×D₄×C₆₇	C₂³	1	1	yes	characteristic
91	#117×C₆₇	—	2	1	no	—
92	#117×C₆₇	—	2	1	no	—
93	#117×C₆₇	C₂×C₄	1	1	yes	characteristic
94	#117×C₆₇	C₂×C₄	1	1	yes	characteristic
95	C₂×C₄²×C₆₇	—	2	1	no	—
96	C₂×C₄²×C₆₇	—	2	1	no	—
97	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
98	#107×C₆₇	C₂×C₄	1	1	yes	characteristic
99	#107×C₆₇	C₂×C₄	1	1	yes	characteristic
100	#106×C₆₇	C₂×C₄	1	1	yes	characteristic
101	#106×C₆₇	C₂×C₄	1	1	yes	characteristic
102	#103×C₆₇	C₂×C₄	1	1	yes	characteristic
103	#103×C₆₇	C₂×C₄	1	1	yes	characteristic
104	#103×C₆₇	C₂×C₄	1	1	yes	characteristic
105	#103×C₆₇	C₂×C₄	1	1	yes	characteristic
106	#102×C₆₇	C₂×C₄	1	1	yes	characteristic
107	#102×C₆₇	C₂×C₄	1	1	yes	characteristic
108	#102×C₆₇	C₂×C₄	1	1	yes	characteristic
109	#102×C₆₇	C₂×C₄	1	1	yes	characteristic
110	C₄×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
111	C₄×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
112	C₄×C₈×C₆₇	C₂³	1	1	yes	characteristic
113	C₄×C₈×C₆₇	C₂³	1	1	yes	characteristic
114	#11761612	—	4	4	no	—
115	C₂×C₆₇⋊C₈	—	2	4	no	—
116	C₂×C₆₇⋊C₈	—	2	2	no	—
117	C₂×C₆₇⋊C₈	—	2	2	no	—
118	C₂×C₆₇⋊C₈	—	2	2	no	—
119	C₂×C₆₇⋊C₈	—	4	2	no	—
120	C₂×C₆₇⋊C₈	—	4	4	no	—
121	C₂×C₆₇⋊C₈	C₂×D₄	1	2	yes	—
122	C₂×C₆₇⋊C₈	C₂×D₄	1	2	yes	—
123	#41×C₆₇	—	2	2	no	—
124	#41×C₆₇	—	2	2	no	—
125	#41×C₆₇	—	2	2	no	—
126	#41×C₆₇	M₁₆	1	4	yes	complemented
127	C₂×Q₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
128	C₂×Q₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
129	C₂×D₄×C₆₇	—	2	1	no	—
130	C₂×D₄×C₆₇	—	2	1	no	—
131	C₂×D₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
132	C₂×D₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
133	C₂²×C₄×C₆₇	—	2	1	no	—
134	C₂²×C₄×C₆₇	—	2	1	no	—
135	C₂²×C₄×C₆₇	—	2	1	no	—
136	C₂²×C₄×C₆₇	—	2	1	no	—
137	C₂²×C₄×C₆₇	—	2	1	no	—
138	C₂²×C₄×C₆₇	—	2	1	no	—
139	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
140	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
141	C₂²×C₄×C₆₇	#41	1	1	yes	characteristic
142	Q₁₆×C₆₇	—	2	2	no	—
143	SD₁₆×C₆₇	—	2	2	no	—
144	SD₁₆×C₆₇	—	2	2	no	—
145	D₈×C₆₇	—	2	2	no	—
146	C₂×C₈×C₆₇	—	2	2	no	—
147	C₂×C₈×C₆₇	—	2	2	no	—
148	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
149	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
150	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
151	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
152	C₄⋊C₄×C₆₇	—	2	1	no	—
153	C₄⋊C₄×C₆₇	—	2	1	no	—
154	C₄⋊C₄×C₆₇	—	2	1	no	—
155	C₄⋊C₄×C₆₇	—	2	1	no	—
156	C₄⋊C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
157	C₄⋊C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
158	C₄⋊C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
159	C₄⋊C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
160	C₂²⋊C₄×C₆₇	—	2	1	no	—
161	C₂²⋊C₄×C₆₇	—	2	1	no	—
162	C₂²⋊C₄×C₆₇	—	2	1	no	—
163	C₂²⋊C₄×C₆₇	—	2	1	no	—
164	C₂²⋊C₄×C₆₇	—	2	1	no	—
165	C₂²⋊C₄×C₆₇	—	2	1	no	—
166	C₄²×C₆₇	—	2	1	no	—
167	C₄²×C₆₇	—	2	1	no	—
168	C₄²×C₆₇	—	2	1	no	—
169	C₄²×C₆₇	—	2	1	no	—
170	C₄²×C₆₇	—	2	1	no	—
171	C₄²×C₆₇	—	2	1	no	—
172	C₄²×C₆₇	C₂²×C₄	1	1	yes	characteristic
173	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
174	C₄²×C₆₇	#41	1	1	yes	characteristic
175	C₄²×C₆₇	C₂⁴	1	1	yes	characteristic
176	C₂³×C₆₇	—	2	1	no	—
177	C₂³×C₆₇	—	2	1	no	—
178	C₂³×C₆₇	C₄×D₄	1	1	yes	characteristic
179	Q₈×C₆₇	—	2	1	no	—
180	Q₈×C₆₇	—	2	1	no	—
181	Q₈×C₆₇	—	2	1	no	—
182	Q₈×C₆₇	C₂×M₁₆	1	2	yes	—
183	D₄×C₆₇	—	2	4	no	—
184	D₄×C₆₇	—	2	1	no	—
185	D₄×C₆₇	—	2	1	no	—
186	D₄×C₆₇	—	2	1	no	—
187	D₄×C₆₇	—	4	2	no	—
188	D₄×C₆₇	C₂×M₁₆	1	2	yes	—
189	C₂×C₄×C₆₇	—	2	1	no	—
190	C₂×C₄×C₆₇	—	2	2	no	—
191	C₂×C₄×C₆₇	—	2	4	no	—
192	C₂×C₄×C₆₇	—	2	2	no	—
193	C₂×C₄×C₆₇	—	2	1	no	—
194	C₂×C₄×C₆₇	—	2	1	no	—
195	C₂×C₄×C₆₇	—	2	1	no	—
196	C₂×C₄×C₆₇	—	2	1	no	—
197	C₂×C₄×C₆₇	—	2	1	no	—
198	C₂×C₄×C₆₇	—	4	2	no	—
199	C₂×C₄×C₆₇	—	4	2	no	—
200	C₂×C₄×C₆₇	—	4	2	no	—
201	C₂×C₄×C₆₇	—	4	2	no	—
202	C₂×C₄×C₆₇	—	4	2	no	—
203	C₂×C₄×C₆₇	C₄×D₄	1	1	yes	characteristic
204	C₂×C₄×C₆₇	C₄×D₄	1	1	yes	characteristic
205	C₂×C₄×C₆₇	C₄×D₄	1	1	yes	characteristic
206	C₂×C₄×C₆₇	C₂×M₁₆	1	2	yes	—
207	C₂×C₄×C₆₇	#131	1	2	yes	—
208	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
209	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
210	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
211	C₈×C₆₇	—	2	1	no	—
212	C₈×C₆₇	—	2	1	no	—
213	C₈×C₆₇	—	2	1	no	—
214	C₈×C₆₇	—	2	1	no	—
215	C₆₇⋊C₈	—	2	2	no	—
216	C₆₇⋊C₈	—	4	2	no	—
217	C₆₇⋊C₈	—	4	2	no	—
218	C₆₇⋊C₈	—	4	2	no	—
219	C₆₇⋊C₈	#135	1	4	yes	complemented
220	C₂²×C₆₇	—	2	1	no	—
221	C₂²×C₆₇	—	2	1	no	—
222	C₂²×C₆₇	—	2	2	no	—
223	C₂²×C₆₇	—	4	1	no	—
224	C₂²×C₆₇	—	4	1	no	—
225	C₂²×C₆₇	—	4	1	no	—
226	C₂²×C₆₇	—	4	1	no	—
227	C₂²×C₆₇	—	4	1	no	—
228	C₂²×C₆₇	#435	1	2	yes	—
229	C₂²×C₆₇	C₂×C₄×D₄	1	1	yes	characteristic
230	C₄×C₆₇	—	2	2	no	—
231	C₄×C₆₇	—	2	1	no	—
232	C₄×C₆₇	—	2	1	no	—
233	C₄×C₆₇	—	4	1	no	—
234	C₄×C₆₇	—	4	1	no	—
235	C₄×C₆₇	—	4	1	no	—
236	C₄×C₆₇	—	4	1	no	—
237	C₄×C₆₇	—	4	1	no	—
238	C₄×C₆₇	#435	1	2	yes	—
239	C₄×C₆₇	#444	1	2	yes	—
240	C₄×C₆₇	C₂²×M₁₆	1	1	yes	characteristic
241	C₄×C₆₇	C₂×#131	1	1	yes	characteristic
242	C₄×C₆₇	C₂×#135	1	2	yes	—
243	#17863	—	67	1	no	maximal, Sylow
244	C₂×C₆₇	—	2	2	no	—
245	C₂×C₆₇	—	4	2	no	—
246	C₂×C₆₇	—	8	1	no	—
247	C₂×C₆₇	C₂×#435	1	1	yes	characteristic
248	C₂×C₆₇	C₄×#135	1	1	yes	characteristic
249	C₂×C₆₇	C₂×#444	1	1	yes	characteristic
250	#2965	—	67	1	no	—
251	C₈×#41	—	67	1	no	—
252	C₄×#135	D₆₇	1	1	yes	characteristic
253	#1584	—	67	2	no	—
254	#1583	—	67	2	no	—
255	#1582	—	67	2	no	—
256	#1581	—	67	2	no	—
257	#1564	—	67	1	no	—
258	#1476	—	67	1	no	—
259	#1476	—	67	1	no	—
260	C₂×#322	—	67	1	no	—
261	C₆₇	#17863	1	1	yes	characteristic, complemented, Sylow
262	C₂×#135	Dic₆₇	1	1	yes	characteristic
263	C₄×#41	C₂×D₆₇	1	1	yes	characteristic
264	C₄×#41	C₂×D₆₇	1	1	yes	characteristic
265	#446	—	67	2	no	—
266	C₈×Q₈	—	67	2	no	—
267	C₄×Q₁₆	Dic₆₇	1	1	yes	characteristic
268	C₄×Q₁₆	C₂×D₆₇	1	1	yes	characteristic
269	C₄×SD₁₆	Dic₆₇	1	1	yes	characteristic
270	C₄×SD₁₆	Dic₆₇	1	1	yes	characteristic
271	C₄×SD₁₆	C₂×D₆₇	1	1	yes	characteristic
272	C₄×SD₁₆	C₂×D₆₇	1	1	yes	characteristic
273	C₄×D₈	Dic₆₇	1	1	yes	characteristic
274	C₄×D₈	C₂×D₆₇	1	1	yes	characteristic
275	#436	—	67	2	no	—
276	#435	—	134	2	no	—
277	C₈×D₄	—	67	2	no	—
278	C₈×D₄	—	134	2	no	—
279	#431	—	67	1	no	—
280	#431	—	67	1	no	—
281	#431	—	134	1	no	—
282	#431	—	134	1	no	—
283	#427	Dic₆₇	1	1	yes	characteristic
284	#416	Dic₆₇	1	1	yes	characteristic
285	#416	Dic₆₇	1	1	yes	characteristic
286	C₄×M₁₆	—	134	1	no	—
287	C₂×C₄×C₈	—	67	1	no	—
288	C₂×C₄×C₈	—	67	1	no	—
289	C₂×C₄×C₈	—	134	1	no	—
290	C₂×C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
291	#335	—	67	2	no	—
292	#334	—	67	2	no	—
293	Q₈⋊C₈	—	67	2	no	—
294	Q₈⋊C₈	—	67	2	no	—
295	D₄⋊C₈	—	67	2	no	—
296	D₄⋊C₈	—	67	2	no	—
297	#322	—	67	2	no	—
298	#322	—	67	2	no	—
299	C₂×#41	C₂×Dic₆₇	1	1	yes	characteristic
300	C₂×#41	C₂×Dic₆₇	1	1	yes	characteristic
301	#135	—	2	4	no	—
302	C₂×Q₁₆	C₂×Dic₆₇	1	1	yes	characteristic
303	C₂×SD₁₆	C₂×Dic₆₇	1	1	yes	characteristic
304	C₂×SD₁₆	C₂×Dic₆₇	1	1	yes	characteristic
305	C₂×D₈	C₂×Dic₆₇	1	1	yes	characteristic
306	C₂×M₁₆	—	134	2	no	—
307	C₂²×C₈	—	67	2	no	—
308	C₂²×C₈	—	134	2	no	—
309	C₂²×C₈	—	134	1	no	—
310	C₂²×C₈	C₂×Dic₆₇	1	1	yes	characteristic
311	C₂²×C₈	C₂×Dic₆₇	1	1	yes	characteristic
312	C₄×Q₈	C₂×Dic₆₇	1	1	yes	characteristic
313	C₄×Q₈	C₂×Dic₆₇	1	1	yes	characteristic
314	C₄×Q₈	C₂²×D₆₇	1	1	yes	characteristic
315	C₄×Q₈	C₂²×D₆₇	1	1	yes	characteristic
316	C₄×D₄	—	2	1	no	—
317	C₄×D₄	—	2	1	no	—
318	C₄×D₄	—	2	1	no	—
319	C₄×D₄	—	2	1	no	—
320	C₄×D₄	C₂×Dic₆₇	1	1	yes	characteristic
321	C₄×D₄	C₂×Dic₆₇	1	1	yes	characteristic
322	C₄×D₄	C₂²×D₆₇	1	1	yes	characteristic
323	C₄×D₄	C₂²×D₆₇	1	1	yes	characteristic
324	#117	—	2	1	no	—
325	#117	—	2	1	no	—
326	#117	C₂×Dic₆₇	1	1	yes	characteristic
327	#117	C₂×Dic₆₇	1	1	yes	characteristic
328	C₂×C₄²	—	2	1	no	—
329	C₂×C₄²	—	2	1	no	—
330	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
331	#107	C₂×Dic₆₇	1	1	yes	characteristic
332	#107	C₂×Dic₆₇	1	1	yes	characteristic
333	#106	C₂×Dic₆₇	1	1	yes	characteristic
334	#106	C₂×Dic₆₇	1	1	yes	characteristic
335	C₄⋊C₈	—	67	2	no	—
336	C₄⋊C₈	—	67	2	no	—
337	C₄⋊C₈	—	134	2	no	—
338	C₄⋊C₈	—	134	2	no	—
339	#103	C₂×Dic₆₇	1	1	yes	characteristic
340	#103	C₂×Dic₆₇	1	1	yes	characteristic
341	#103	C₂×Dic₆₇	1	1	yes	characteristic
342	#103	C₂×Dic₆₇	1	1	yes	characteristic
343	#102	C₂×Dic₆₇	1	1	yes	characteristic
344	#102	C₂×Dic₆₇	1	1	yes	characteristic
345	#102	C₂×Dic₆₇	1	1	yes	characteristic
346	#102	C₂×Dic₆₇	1	1	yes	characteristic
347	C₂²⋊C₈	—	134	2	no	—
348	C₂²⋊C₈	—	134	2	no	—
349	C₂²⋊C₈	—	134	1	no	—
350	C₂²⋊C₈	—	134	1	no	—
351	#97	—	134	2	no	—
352	C₄×C₈	—	67	1	no	—
353	C₄×C₈	—	67	1	no	—
354	C₄×C₈	—	67	2	no	—
355	C₄×C₈	—	67	2	no	—
356	C₄×C₈	—	134	2	no	—
357	C₄×C₈	—	134	1	no	—
358	C₄×C₈	—	134	1	no	—
359	C₄×C₈	—	134	1	no	—
360	C₄×C₈	C₂×Dic₆₇	1	1	yes	characteristic
361	C₄×C₈	C₂×Dic₆₇	1	1	yes	characteristic
362	C₄×C₈	C₂²×D₆₇	1	1	yes	characteristic
363	C₄×C₈	C₂²×D₆₇	1	1	yes	characteristic
364	#41	—	2	2	no	—
365	#41	—	2	2	no	—
366	#41	—	2	2	no	—
367	#41	#11761612	1	4	yes	complemented
368	C₂×Q₈	C₂²×Dic₆₇	1	1	yes	characteristic
369	C₂×Q₈	C₂²×Dic₆₇	1	1	yes	characteristic
370	C₂×D₄	—	2	1	no	—
371	C₂×D₄	—	2	1	no	—
372	C₂×D₄	C₂²×Dic₆₇	1	1	yes	characteristic
373	C₂×D₄	C₂²×Dic₆₇	1	1	yes	characteristic
374	C₂²×C₄	—	2	1	no	—
375	C₂²×C₄	—	2	1	no	—
376	C₂²×C₄	—	2	1	no	—
377	C₂²×C₄	—	2	1	no	—
378	C₂²×C₄	—	2	1	no	—
379	C₂²×C₄	—	2	1	no	—
380	C₂²×C₄	C₂²×Dic₆₇	1	1	yes	characteristic
381	C₂²×C₄	D₄×D₆₇	1	1	yes	characteristic
382	C₂²×C₄	#11761627	1	1	yes	characteristic
383	Q₁₆	—	2	2	no	—
384	SD₁₆	—	2	2	no	—
385	SD₁₆	—	2	2	no	—
386	D₈	—	2	2	no	—
387	M₁₆	—	268	4	no	—
388	C₂×C₈	—	2	2	no	—
389	C₂×C₈	—	2	2	no	—
390	C₂×C₈	—	268	2	no	—
391	C₂×C₈	—	268	4	no	—
392	C₂×C₈	—	67	2	no	—
393	C₂×C₈	—	67	2	no	—
394	C₂×C₈	—	134	2	no	—
395	C₂×C₈	—	134	2	no	—
396	C₂×C₈	—	134	2	no	—
397	C₂×C₈	—	134	4	no	—
398	C₂×C₈	C₂²×Dic₆₇	1	1	yes	characteristic
399	C₂×C₈	C₂²×Dic₆₇	1	1	yes	characteristic
400	C₂×C₈	C₂²×Dic₆₇	1	1	yes	characteristic
401	C₂×C₈	C₂²×Dic₆₇	1	1	yes	characteristic
402	C₄⋊C₄	—	2	1	no	—
403	C₄⋊C₄	—	2	1	no	—
404	C₄⋊C₄	—	2	1	no	—
405	C₄⋊C₄	—	2	1	no	—
406	C₄⋊C₄	C₂²×Dic₆₇	1	1	yes	characteristic
407	C₄⋊C₄	C₂²×Dic₆₇	1	1	yes	characteristic
408	C₄⋊C₄	C₂²×Dic₆₇	1	1	yes	characteristic
409	C₄⋊C₄	C₂²×Dic₆₇	1	1	yes	characteristic
410	C₂²⋊C₄	—	2	1	no	—
411	C₂²⋊C₄	—	2	1	no	—
412	C₂²⋊C₄	—	2	1	no	—
413	C₂²⋊C₄	—	2	1	no	—
414	C₂²⋊C₄	—	2	1	no	—
415	C₂²⋊C₄	—	2	1	no	—
416	C₄²	—	2	1	no	—
417	C₄²	—	2	1	no	—
418	C₄²	—	2	1	no	—
419	C₄²	—	2	1	no	—
420	C₄²	—	2	1	no	—
421	C₄²	—	2	1	no	—
422	C₄²	C₂²×Dic₆₇	1	1	yes	characteristic
423	C₄²	D₄×D₆₇	1	1	yes	characteristic
424	C₄²	#11761627	1	1	yes	characteristic
425	C₄²	C₂³×D₆₇	1	1	yes	characteristic
426	C₂³	—	2	1	no	—
427	C₂³	—	2	1	no	—
428	C₂³	D₄×Dic₆₇	1	1	yes	characteristic
429	Q₈	—	2	1	no	—
430	Q₈	—	2	1	no	—
431	Q₈	—	2	1	no	—
432	Q₈	C₂×#11761612	1	2	yes	—
433	D₄	—	2	4	no	—
434	D₄	—	2	1	no	—
435	D₄	—	2	1	no	—
436	D₄	—	2	1	no	—
437	D₄	—	4	2	no	—
438	D₄	C₂×#11761612	1	2	yes	—
439	C₂×C₄	—	2	1	no	—
440	C₂×C₄	—	2	2	no	—
441	C₂×C₄	—	2	2	no	—
442	C₂×C₄	—	2	4	no	—
443	C₂×C₄	—	2	1	no	—
444	C₂×C₄	—	2	1	no	—
445	C₂×C₄	—	2	1	no	—
446	C₂×C₄	—	2	1	no	—
447	C₂×C₄	—	2	1	no	—
448	C₂×C₄	—	4	2	no	—
449	C₂×C₄	—	4	2	no	—
450	C₂×C₄	—	4	2	no	—
451	C₂×C₄	—	4	2	no	—
452	C₂×C₄	—	4	2	no	—
453	C₂×C₄	D₄×Dic₆₇	1	1	yes	characteristic
454	C₂×C₄	D₄×Dic₆₇	1	1	yes	characteristic
455	C₂×C₄	D₄×Dic₆₇	1	1	yes	characteristic
456	C₂×C₄	C₂×#11761612	1	2	yes	—
457	C₂×C₄	#14524050	1	2	yes	—
458	C₂×C₄	C₂³×Dic₆₇	1	1	yes	characteristic
459	C₂×C₄	C₂×D₄×D₆₇	1	1	yes	characteristic
460	C₂×C₄	C₂×#11761627	1	1	yes	characteristic
461	C₈	—	2	1	no	—
462	C₈	—	2	1	no	—
463	C₈	—	2	1	no	—
464	C₈	—	2	1	no	—
465	C₈	—	268	2	no	—
466	C₈	—	268	2	no	—
467	C₈	—	268	2	no	—
468	C₈	—	67	4	no	—
469	C₈	—	134	2	no	—
470	C₂²	—	2	1	no	—
471	C₂²	—	2	1	no	—
472	C₂²	—	2	2	no	—
473	C₂²	—	4	1	no	—
474	C₂²	—	4	1	no	—
475	C₂²	—	4	1	no	—
476	C₂²	—	4	1	no	—
477	C₂²	—	4	1	no	—
478	C₂²	#16863468	1	2	yes	—
479	C₂²	C₂×D₄×Dic₆₇	1	1	yes	characteristic
480	C₄	—	2	1	no	—
481	C₄	—	2	1	no	—
482	C₄	—	2	2	no	—
483	C₄	—	4	1	no	—
484	C₄	—	4	1	no	—
485	C₄	—	4	1	no	—
486	C₄	—	4	1	no	—
487	C₄	—	4	1	no	—
488	C₄	#16863468	1	2	yes	—
489	C₄	#16863518	1	2	yes	—
490	C₄	C₂²×#11761612	1	1	yes	characteristic
491	C₄	C₂×#14524050	1	1	yes	characteristic
492	C₄	#135×D₆₇	1	2	yes	—
493	C₂	—	2	2	no	—
494	C₂	—	4	2	no	—
495	C₂	—	8	1	no	—
496	C₂	C₂×#16863468	1	1	yes	characteristic
497	C₂	#135×Dic₆₇	1	1	yes	characteristic
498	C₂	C₂×#16863518	1	1	yes	characteristic

	subgroup	G/H
center	C₄²	D₄×D₆₇
commutator	C₂×C₄×C₆₇	C₂³×C₄
Frattini	C₄²	C₂³×D₆₇
Fitting	C₄×#135×C₆₇	C₂
socle	C₂²×C₆₇	C₂×C₄×D₄
prime radical	C₂×#135×C₆₇	C₄
cosocle	C₄²×C₆₇	C₂⁴
hypercenter	C₄×#135	D₆₇
hypocenter	C₆₇	#17863
LCS	G ▷ C₂×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₄² ◁ C₂×C₄² ◁ C₄×#135

Complements

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

N	H ≅ G/N	type
1. #22569176	C₂	semidirect
2. #41×C₆₇⋊C₈	C₂	semidirect
3. #22559060	C₂	semidirect
4. #22559053	C₂	semidirect
5. #22559046	C₂	semidirect
6. #22559039	C₂	semidirect
7. #22558957	C₂	semidirect
8. #22558591	C₂	semidirect
9. #22558591	C₂	semidirect
10. C₂×#16863118	C₂	semidirect
126. #41×C₆₇	M₁₆	semidirect
219. C₆₇⋊C₈	#135	semidirect
261. C₆₇	#17863	semidirect
367. #41	#11761612	semidirect

Maximal subgroups

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

subgroup	index
#22569176	2
#41×C₆₇⋊C₈	2
#22559060	2
#22559053	2
#22559046	2
#22559039	2
#22558957	2
#22558591	2
C₂×#16863118	2
C₄×#135×C₆₇	2
#17863	67