Subgroups — C₆₇×#22418 #30701669

Order 17152.

1542 conjugacy classes of subgroups.

The subgroup lattice has 596 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#550×C₆₇	C₂	1	1	yes	maximal, characteristic
2	C₂×C₈×D₄×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#416×C₆₇	C₂	1	2	yes	maximal, complemented
4	C₆₇×#2354	C₂	1	1	yes	maximal, characteristic, complemented
5	C₆₇×#1982	C₂	1	1	yes	maximal, characteristic, complemented
6	C₆₇×#1957	C₂	1	1	yes	maximal, characteristic, complemented
7	C₆₇×#1893	C₂	1	2	yes	maximal, complemented
8	C₆₇×#1875	C₂	1	2	yes	maximal, complemented
9	C₆₇×#1803	C₂	1	1	yes	maximal, characteristic, complemented
10	C₆₇×#1788	C₂	1	2	yes	maximal, complemented
11	C₄×#103×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂²×#41×C₆₇	C₂²	1	2	yes	—
13	C₂³×C₈×C₆₇	C₂²	1	2	yes	—
14	#550×C₆₇	—	2	4	no	—
15	#550×C₆₇	C₄	1	8	yes	complemented
16	C₂×#127×C₆₇	—	2	1	no	—
17	C₂×#127×C₆₇	C₂²	1	1	yes	characteristic
18	C₂×#121×C₆₇	—	2	2	no	—
19	C₂×#121×C₆₇	C₂²	1	2	yes	—
20	C₂×C₄×Q₈×C₆₇	C₂²	1	1	yes	characteristic
21	C₂×C₄×D₄×C₆₇	—	2	1	no	—
22	C₂×C₄×D₄×C₆₇	C₂²	1	1	yes	characteristic
23	C₂×#117×C₆₇	C₂²	1	2	yes	—
24	C₈×D₄×C₆₇	—	2	4	no	—
25	C₂×C₄⋊C₈×C₆₇	C₂²	1	1	yes	characteristic
26	#416×C₆₇	—	2	8	no	—
27	C₂×#103×C₆₇	C₂²	1	1	yes	characteristic
28	C₂×#103×C₆₇	C₂²	1	2	yes	—
29	C₂×#103×C₆₇	C₂²	1	1	yes	characteristic
30	C₂×#102×C₆₇	C₂²	1	2	yes	—
31	C₂×#102×C₆₇	C₂²	1	2	yes	—
32	C₂×#102×C₆₇	C₂²	1	2	yes	—
33	C₂×#102×C₆₇	C₂²	1	2	yes	—
34	C₂×C₂²⋊C₈×C₆₇	C₂²	1	2	yes	—
35	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
36	#390×C₆₇	—	2	1	no	—
37	#388×C₆₇	—	2	2	no	—
38	#384×C₆₇	C₂²	1	1	yes	characteristic
39	#380×C₆₇	C₂²	1	2	yes	—
40	C₄×C₄⋊C₄×C₆₇	—	2	1	no	—
41	C₄×C₄⋊C₄×C₆₇	C₂²	1	1	yes	characteristic
42	#340×C₆₇	C₂²	1	1	yes	characteristic
43	#340×C₆₇	C₂²	1	1	yes	characteristic
44	#340×C₆₇	C₂²	1	2	yes	—
45	C₂×#41×C₆₇	—	2	4	no	—
46	C₂×#41×C₆₇	—	2	4	no	—
47	C₂×#41×C₆₇	—	2	4	no	—
48	C₂×#41×C₆₇	—	4	2	no	—
49	C₂×#41×C₆₇	—	4	2	no	—
50	C₂×#41×C₆₇	C₂×C₄	1	8	yes	—
51	C₂²×Q₈×C₆₇	C₂³	1	1	yes	characteristic
52	C₂²×D₄×C₆₇	—	2	2	no	—
53	C₂²×D₄×C₆₇	—	2	2	no	—
54	C₂²×D₄×C₆₇	—	2	1	no	—
55	C₂²×D₄×C₆₇	—	2	1	no	—
56	C₂²×D₄×C₆₇	D₄	1	1	yes	characteristic
57	C₂²×D₄×C₆₇	C₂³	1	2	yes	—
58	C₂³×C₄×C₆₇	—	2	2	no	—
59	C₂³×C₄×C₆₇	C₂³	1	2	yes	—
60	C₂²×C₈×C₆₇	—	2	4	no	—
61	C₂²×C₈×C₆₇	—	2	4	no	—
62	C₂²×C₈×C₆₇	—	4	2	no	—
63	C₂²×C₈×C₆₇	C₂³	1	2	yes	—
64	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
65	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
66	#127×C₆₇	—	2	8	no	—
67	#127×C₆₇	—	2	2	no	—
68	#127×C₆₇	C₂×C₄	1	4	yes	—
69	#121×C₆₇	—	2	4	no	—
70	#121×C₆₇	—	2	16	no	—
71	#121×C₆₇	C₂×C₄	1	8	yes	—
72	C₄×Q₈×C₆₇	—	2	2	no	—
73	C₄×Q₈×C₆₇	C₂×C₄	1	4	yes	—
74	C₄×D₄×C₆₇	—	2	8	no	—
75	C₄×D₄×C₆₇	C₂×C₄	1	4	yes	—
76	C₄×D₄×C₆₇	D₄	1	4	yes	—
77	#117×C₆₇	—	2	4	no	—
78	#117×C₆₇	—	4	2	no	—
79	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
80	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
81	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
82	C₂×C₄⋊C₄×C₆₇	D₄	1	1	yes	characteristic
83	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
84	C₂×C₄⋊C₄×C₆₇	C₂³	1	1	yes	characteristic
85	C₂×C₄⋊C₄×C₆₇	C₂³	1	1	yes	characteristic
86	C₂×C₄⋊C₄×C₆₇	C₂³	1	1	yes	characteristic
87	C₂×C₂²⋊C₄×C₆₇	—	2	2	no	—
88	C₂×C₂²⋊C₄×C₆₇	—	2	2	no	—
89	C₂×C₂²⋊C₄×C₆₇	—	2	2	no	—
90	C₂×C₂²⋊C₄×C₆₇	D₄	1	2	yes	—
91	C₂×C₄²×C₆₇	—	2	1	no	—
92	C₂×C₄²×C₆₇	—	2	2	no	—
93	C₂×C₄²×C₆₇	—	2	1	no	—
94	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
95	C₄⋊C₈×C₆₇	—	2	2	no	—
96	#103×C₆₇	—	2	4	no	—
97	#103×C₆₇	—	2	2	no	—
98	#103×C₆₇	D₄	1	4	yes	complemented
99	#102×C₆₇	—	2	4	no	—
100	#102×C₆₇	—	2	4	no	—
101	#102×C₆₇	—	4	2	no	—
102	#102×C₆₇	—	4	2	no	—
103	C₂²⋊C₈×C₆₇	—	2	4	no	—
104	C₄×C₈×C₆₇	—	2	2	no	—
105	#95×C₆₇	—	2	2	no	—
106	#95×C₆₇	—	2	1	no	—
107	C₂⁴×C₆₇	—	2	2	no	—
108	C₂⁴×C₆₇	C₂×D₄	1	2	yes	—
109	#41×C₆₇	—	2	16	no	—
110	#41×C₆₇	—	4	8	no	—
111	#41×C₆₇	—	4	8	no	—
112	#41×C₆₇	—	4	8	no	—
113	C₂×Q₈×C₆₇	—	2	1	no	—
114	C₂×Q₈×C₆₇	—	2	1	no	—
115	C₂×Q₈×C₆₇	—	2	1	no	—
116	C₂×Q₈×C₆₇	C₂²×C₄	1	2	yes	—
117	C₂×Q₈×C₆₇	C₂×D₄	1	2	yes	—
118	C₂×Q₈×C₆₇	#41	1	2	yes	—
119	C₂×D₄×C₆₇	—	2	8	no	—
120	C₂×D₄×C₆₇	—	2	8	no	—
121	C₂×D₄×C₆₇	—	2	4	no	—
122	C₂×D₄×C₆₇	—	2	4	no	—
123	C₂×D₄×C₆₇	—	2	2	no	—
124	C₂×D₄×C₆₇	—	2	2	no	—
125	C₂×D₄×C₆₇	—	2	2	no	—
126	C₂×D₄×C₆₇	—	2	2	no	—
127	C₂×D₄×C₆₇	—	2	2	no	—
128	C₂×D₄×C₆₇	—	2	2	no	—
129	C₂×D₄×C₆₇	—	2	2	no	—
130	C₂×D₄×C₆₇	—	4	4	no	—
131	C₂×D₄×C₆₇	—	4	4	no	—
132	C₂×D₄×C₆₇	—	4	2	no	—
133	C₂×D₄×C₆₇	—	4	2	no	—
134	C₂×D₄×C₆₇	—	8	1	no	—
135	C₂×D₄×C₆₇	—	8	1	no	—
136	C₂×D₄×C₆₇	—	8	2	no	—
137	C₂×D₄×C₆₇	—	8	2	no	—
138	C₂×D₄×C₆₇	C₂²⋊C₄	1	4	yes	—
139	C₂×D₄×C₆₇	C₂²×C₄	1	4	yes	—
140	C₂²×C₄×C₆₇	—	2	1	no	—
141	C₂²×C₄×C₆₇	—	2	2	no	—
142	C₂²×C₄×C₆₇	—	2	2	no	—
143	C₂²×C₄×C₆₇	—	2	2	no	—
144	C₂²×C₄×C₆₇	—	2	2	no	—
145	C₂²×C₄×C₆₇	—	2	1	no	—
146	C₂²×C₄×C₆₇	—	2	1	no	—
147	C₂²×C₄×C₆₇	—	2	8	no	—
148	C₂²×C₄×C₆₇	—	2	2	no	—
149	C₂²×C₄×C₆₇	—	2	1	no	—
150	C₂²×C₄×C₆₇	—	4	4	no	—
151	C₂²×C₄×C₆₇	—	4	4	no	—
152	C₂²×C₄×C₆₇	C₂²×C₄	1	4	yes	—
153	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
154	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
155	C₂²×C₄×C₆₇	C₂×D₄	1	4	yes	—
156	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
157	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
158	C₂×C₈×C₆₇	—	2	1	no	—
159	C₂×C₈×C₆₇	—	2	1	no	—
160	C₂×C₈×C₆₇	—	2	2	no	—
161	C₂×C₈×C₆₇	—	2	2	no	—
162	C₂×C₈×C₆₇	—	2	1	no	—
163	C₂×C₈×C₆₇	—	2	1	no	—
164	C₂×C₈×C₆₇	—	2	1	no	—
165	C₂×C₈×C₆₇	—	2	2	no	—
166	C₂×C₈×C₆₇	—	4	8	no	—
167	C₂×C₈×C₆₇	C₂×D₄	1	2	yes	—
168	C₄⋊C₄×C₆₇	—	2	4	no	—
169	C₄⋊C₄×C₆₇	—	2	1	no	—
170	C₄⋊C₄×C₆₇	—	2	1	no	—
171	C₄⋊C₄×C₆₇	—	2	2	no	—
172	C₄⋊C₄×C₆₇	—	2	1	no	—
173	C₄⋊C₄×C₆₇	—	2	1	no	—
174	C₄⋊C₄×C₆₇	—	2	2	no	—
175	C₄⋊C₄×C₆₇	—	4	2	no	—
176	C₄⋊C₄×C₆₇	—	8	1	no	—
177	C₄⋊C₄×C₆₇	C₂²⋊C₄	1	4	yes	—
178	C₄⋊C₄×C₆₇	C₂²×C₄	1	2	yes	—
179	C₄⋊C₄×C₆₇	C₂×D₄	1	2	yes	—
180	C₂²⋊C₄×C₆₇	—	2	8	no	—
181	C₂²⋊C₄×C₆₇	—	4	4	no	—
182	C₂²⋊C₄×C₆₇	—	8	2	no	—
183	C₂²⋊C₄×C₆₇	C₂²⋊C₄	1	8	yes	—
184	C₄²×C₆₇	—	2	4	no	—
185	C₄²×C₆₇	—	4	4	no	—
186	C₄²×C₆₇	—	4	2	no	—
187	C₄²×C₆₇	C₂²×C₄	1	2	yes	—
188	C₄²×C₆₇	C₂×D₄	1	2	yes	—
189	C₂³×C₆₇	—	2	2	no	—
190	C₂³×C₆₇	—	2	2	no	—
191	C₂³×C₆₇	—	2	2	no	—
192	C₂³×C₆₇	—	2	2	no	—
193	C₂³×C₆₇	—	2	2	no	—
194	C₂³×C₆₇	—	2	2	no	—
195	C₂³×C₆₇	—	2	4	no	—
196	C₂³×C₆₇	—	4	2	no	—
197	C₂³×C₆₇	—	4	2	no	—
198	C₂³×C₆₇	—	4	2	no	—
199	C₂³×C₆₇	—	4	2	no	—
200	C₂³×C₆₇	—	4	2	no	—
201	C₂³×C₆₇	—	4	2	no	—
202	C₂³×C₆₇	C₂×C₂²⋊C₄	1	4	yes	—
203	C₂³×C₆₇	C₂²×D₄	1	1	yes	characteristic
204	Q₈×C₆₇	—	2	2	no	—
205	Q₈×C₆₇	—	2	2	no	—
206	Q₈×C₆₇	—	2	2	no	—
207	Q₈×C₆₇	C₄×D₄	1	4	yes	—
208	D₄×C₆₇	—	2	4	no	—
209	D₄×C₆₇	—	2	4	no	—
210	D₄×C₆₇	—	2	8	no	—
211	D₄×C₆₇	—	4	16	no	—
212	D₄×C₆₇	—	4	2	no	—
213	D₄×C₆₇	—	4	2	no	—
214	D₄×C₆₇	—	4	8	no	—
215	D₄×C₆₇	—	8	4	no	—
216	D₄×C₆₇	—	8	8	no	—
217	C₂×C₄×C₆₇	—	2	2	no	—
218	C₂×C₄×C₆₇	—	2	2	no	—
219	C₂×C₄×C₆₇	—	2	2	no	—
220	C₂×C₄×C₆₇	—	2	2	no	—
221	C₂×C₄×C₆₇	—	2	2	no	—
222	C₂×C₄×C₆₇	—	2	1	no	—
223	C₂×C₄×C₆₇	—	2	1	no	—
224	C₂×C₄×C₆₇	—	2	4	no	—
225	C₂×C₄×C₆₇	—	2	4	no	—
226	C₂×C₄×C₆₇	—	2	4	no	—
227	C₂×C₄×C₆₇	—	2	4	no	—
228	C₂×C₄×C₆₇	—	2	2	no	—
229	C₂×C₄×C₆₇	—	2	2	no	—
230	C₂×C₄×C₆₇	—	4	16	no	—
231	C₂×C₄×C₆₇	—	4	1	no	—
232	C₂×C₄×C₆₇	—	4	1	no	—
233	C₂×C₄×C₆₇	—	4	1	no	—
234	C₂×C₄×C₆₇	—	4	1	no	—
235	C₂×C₄×C₆₇	—	4	2	no	—
236	C₂×C₄×C₆₇	—	4	1	no	—
237	C₂×C₄×C₆₇	—	4	1	no	—
238	C₂×C₄×C₆₇	—	4	2	no	—
239	C₂×C₄×C₆₇	—	4	1	no	—
240	C₂×C₄×C₆₇	—	4	2	no	—
241	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	2	yes	—
242	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	4	yes	—
243	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	2	yes	—
244	C₂×C₄×C₆₇	#135	1	2	yes	—
245	C₂×C₄×C₆₇	#135	1	2	yes	—
246	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
247	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
248	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
249	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
250	C₂×C₄×C₆₇	H(5,𝔽₂)	1	1	yes	characteristic
251	C₂×C₄×C₆₇	H(5,𝔽₂)	1	1	yes	characteristic
252	C₈×C₆₇	—	2	2	no	—
253	C₈×C₆₇	—	4	2	no	—
254	C₈×C₆₇	—	4	1	no	—
255	C₂²×C₆₇	—	2	4	no	—
256	C₂²×C₆₇	—	2	4	no	—
257	C₂²×C₆₇	—	2	4	no	—
258	C₂²×C₆₇	—	4	2	no	—
259	C₂²×C₆₇	—	4	2	no	—
260	C₂²×C₆₇	—	4	2	no	—
261	C₂²×C₆₇	—	4	2	no	—
262	C₂²×C₆₇	—	4	4	no	—
263	C₂²×C₆₇	—	4	4	no	—
264	C₂²×C₆₇	—	4	4	no	—
265	C₂²×C₆₇	—	8	2	no	—
266	C₂²×C₆₇	—	8	2	no	—
267	C₂²×C₆₇	—	8	2	no	—
268	C₂²×C₆₇	—	8	2	no	—
269	C₂²×C₆₇	C₂²×C₂²⋊C₄	1	1	yes	characteristic
270	C₂²×C₆₇	D₄²	1	1	yes	characteristic
271	C₂²×C₆₇	#546	1	1	yes	characteristic
272	C₂²×C₆₇	C₂×#135	1	1	yes	characteristic
273	C₂²×C₆₇	C₂×#135	1	1	yes	characteristic
274	C₂²×C₆₇	#576	1	1	yes	characteristic
275	C₂²×C₆₇	#577	1	1	yes	characteristic
276	C₄×C₆₇	—	2	4	no	—
277	C₄×C₆₇	—	2	4	no	—
278	C₄×C₆₇	—	2	2	no	—
279	C₄×C₆₇	—	4	2	no	—
280	C₄×C₆₇	—	4	2	no	—
281	C₄×C₆₇	—	8	2	no	—
282	C₄×C₆₇	—	8	1	no	—
283	C₄×C₆₇	#416	1	4	yes	complemented
284	C₄×C₆₇	C₂×C₄×D₄	1	1	yes	characteristic
285	C₄×C₆₇	#518	1	1	yes	characteristic
286	C₄×C₆₇	D₄²	1	1	yes	characteristic
287	C₄×C₆₇	#546	1	1	yes	characteristic
288	#22418	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
289	C₂×C₆₇	—	4	4	no	—
290	C₂×C₆₇	—	8	4	no	—
291	C₂×C₆₇	D₄×C₂²⋊C₄	1	1	yes	characteristic
292	C₂×C₆₇	C₂×#416	1	1	yes	characteristic
293	C₂×C₆₇	#2897	1	1	yes	characteristic
294	C₂×C₆₇	#3284	1	1	yes	characteristic
295	C₂×C₆₇	#3287	1	1	yes	characteristic
296	C₂×C₆₇	#3304	1	1	yes	characteristic
297	C₂×C₆₇	#3305	1	1	yes	characteristic
298	C₂×#550	C₂×C₆₇	1	1	yes	characteristic
299	C₂×C₈×D₄	C₂×C₆₇	1	1	yes	characteristic, complemented
300	C₂×#416	C₂×C₆₇	1	2	yes	complemented
301	#2354	C₂×C₆₇	1	1	yes	characteristic, complemented
302	#1982	C₂×C₆₇	1	1	yes	characteristic, complemented
303	#1957	C₂×C₆₇	1	1	yes	characteristic, complemented
304	#1893	C₂×C₆₇	1	2	yes	complemented
305	#1875	C₂×C₆₇	1	2	yes	complemented
306	#1803	C₂×C₆₇	1	1	yes	characteristic, complemented
307	#1788	C₂×C₆₇	1	2	yes	complemented
308	C₄×#103	C₂×C₆₇	1	1	yes	characteristic, complemented
309	C₆₇	#22418	1	1	yes	characteristic, direct product factor, Sylow
310	C₂²×#41	C₂²×C₆₇	1	2	yes	—
311	C₂³×C₈	C₂²×C₆₇	1	2	yes	—
312	#550	—	2	4	no	—
313	#550	C₄×C₆₇	1	8	yes	complemented
314	C₂×#127	—	2	1	no	—
315	C₂×#127	C₂²×C₆₇	1	1	yes	characteristic
316	C₂×#121	—	2	2	no	—
317	C₂×#121	C₂²×C₆₇	1	2	yes	—
318	C₂×C₄×Q₈	C₂²×C₆₇	1	1	yes	characteristic
319	C₂×C₄×D₄	—	2	1	no	—
320	C₂×C₄×D₄	C₂²×C₆₇	1	1	yes	characteristic
321	C₂×#117	C₂²×C₆₇	1	2	yes	—
322	C₈×D₄	—	2	4	no	—
323	C₂×C₄⋊C₈	C₂²×C₆₇	1	1	yes	characteristic
324	#416	—	2	8	no	—
325	C₂×#103	C₂²×C₆₇	1	2	yes	—
326	C₂×#103	C₂²×C₆₇	1	1	yes	characteristic
327	C₂×#103	C₂²×C₆₇	1	1	yes	characteristic
328	C₂×#102	C₂²×C₆₇	1	2	yes	—
329	C₂×#102	C₂²×C₆₇	1	2	yes	—
330	C₂×#102	C₂²×C₆₇	1	2	yes	—
331	C₂×#102	C₂²×C₆₇	1	2	yes	—
332	C₂×C₂²⋊C₈	C₂²×C₆₇	1	2	yes	—
333	C₂×C₄×C₈	C₂²×C₆₇	1	1	yes	characteristic
334	#390	—	2	1	no	—
335	#388	—	2	2	no	—
336	#384	C₂²×C₆₇	1	1	yes	characteristic
337	#380	C₂²×C₆₇	1	2	yes	—
338	C₄×C₄⋊C₄	—	2	1	no	—
339	C₄×C₄⋊C₄	C₂²×C₆₇	1	1	yes	characteristic
340	#340	C₂²×C₆₇	1	2	yes	—
341	#340	C₂²×C₆₇	1	1	yes	characteristic
342	#340	C₂²×C₆₇	1	1	yes	characteristic
343	C₂×#41	—	2	4	no	—
344	C₂×#41	—	2	4	no	—
345	C₂×#41	—	2	4	no	—
346	C₂×#41	—	4	2	no	—
347	C₂×#41	—	4	2	no	—
348	C₂×#41	C₂×C₄×C₆₇	1	8	yes	—
349	C₂²×Q₈	C₂³×C₆₇	1	1	yes	characteristic
350	C₂²×D₄	—	2	2	no	—
351	C₂²×D₄	—	2	2	no	—
352	C₂²×D₄	—	2	1	no	—
353	C₂²×D₄	—	2	1	no	—
354	C₂²×D₄	D₄×C₆₇	1	1	yes	characteristic
355	C₂²×D₄	C₂³×C₆₇	1	2	yes	—
356	C₂³×C₄	—	2	2	no	—
357	C₂³×C₄	C₂³×C₆₇	1	2	yes	—
358	C₂²×C₈	—	2	4	no	—
359	C₂²×C₈	—	2	4	no	—
360	C₂²×C₈	—	4	2	no	—
361	C₂²×C₈	C₂³×C₆₇	1	2	yes	—
362	C₂²×C₈	C₂³×C₆₇	1	1	yes	characteristic
363	C₂²×C₈	C₂³×C₆₇	1	1	yes	characteristic
364	#127	—	2	8	no	—
365	#127	—	2	2	no	—
366	#127	C₂×C₄×C₆₇	1	4	yes	—
367	#121	—	2	4	no	—
368	#121	—	2	16	no	—
369	#121	C₂×C₄×C₆₇	1	8	yes	—
370	C₄×Q₈	—	2	2	no	—
371	C₄×Q₈	C₂×C₄×C₆₇	1	4	yes	—
372	C₄×D₄	—	2	8	no	—
373	C₄×D₄	C₂×C₄×C₆₇	1	4	yes	—
374	C₄×D₄	D₄×C₆₇	1	4	yes	—
375	#117	—	2	4	no	—
376	#117	—	4	2	no	—
377	C₂×C₄⋊C₄	—	2	1	no	—
378	C₂×C₄⋊C₄	—	2	1	no	—
379	C₂×C₄⋊C₄	—	2	1	no	—
380	C₂×C₄⋊C₄	D₄×C₆₇	1	1	yes	characteristic
381	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
382	C₂×C₄⋊C₄	C₂³×C₆₇	1	1	yes	characteristic
383	C₂×C₄⋊C₄	C₂³×C₆₇	1	1	yes	characteristic
384	C₂×C₄⋊C₄	C₂³×C₆₇	1	1	yes	characteristic
385	C₂×C₂²⋊C₄	—	2	2	no	—
386	C₂×C₂²⋊C₄	—	2	2	no	—
387	C₂×C₂²⋊C₄	—	2	2	no	—
388	C₂×C₂²⋊C₄	D₄×C₆₇	1	2	yes	—
389	C₂×C₄²	—	2	1	no	—
390	C₂×C₄²	—	2	2	no	—
391	C₂×C₄²	—	2	1	no	—
392	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
393	C₄⋊C₈	—	2	2	no	—
394	#103	—	2	4	no	—
395	#103	—	2	2	no	—
396	#103	D₄×C₆₇	1	4	yes	complemented
397	#102	—	2	4	no	—
398	#102	—	2	4	no	—
399	#102	—	4	2	no	—
400	#102	—	4	2	no	—
401	C₂²⋊C₈	—	2	4	no	—
402	C₄×C₈	—	2	2	no	—
403	#95	—	2	2	no	—
404	#95	—	2	1	no	—
405	C₂⁴	—	2	2	no	—
406	C₂⁴	C₂×D₄×C₆₇	1	2	yes	—
407	#41	—	2	16	no	—
408	#41	—	4	8	no	—
409	#41	—	4	8	no	—
410	#41	—	4	8	no	—
411	C₂×Q₈	—	2	1	no	—
412	C₂×Q₈	—	2	1	no	—
413	C₂×Q₈	—	2	1	no	—
414	C₂×Q₈	C₂²×C₄×C₆₇	1	2	yes	—
415	C₂×Q₈	C₂×D₄×C₆₇	1	2	yes	—
416	C₂×Q₈	#41×C₆₇	1	2	yes	—
417	C₂×D₄	—	2	8	no	—
418	C₂×D₄	—	2	8	no	—
419	C₂×D₄	—	2	4	no	—
420	C₂×D₄	—	2	4	no	—
421	C₂×D₄	—	2	2	no	—
422	C₂×D₄	—	2	2	no	—
423	C₂×D₄	—	2	2	no	—
424	C₂×D₄	—	2	2	no	—
425	C₂×D₄	—	2	2	no	—
426	C₂×D₄	—	2	2	no	—
427	C₂×D₄	—	2	2	no	—
428	C₂×D₄	—	4	4	no	—
429	C₂×D₄	—	4	4	no	—
430	C₂×D₄	—	4	2	no	—
431	C₂×D₄	—	4	2	no	—
432	C₂×D₄	—	8	1	no	—
433	C₂×D₄	—	8	1	no	—
434	C₂×D₄	—	8	2	no	—
435	C₂×D₄	—	8	2	no	—
436	C₂×D₄	C₂²⋊C₄×C₆₇	1	4	yes	—
437	C₂×D₄	C₂²×C₄×C₆₇	1	4	yes	—
438	C₂²×C₄	—	2	1	no	—
439	C₂²×C₄	—	2	2	no	—
440	C₂²×C₄	—	2	2	no	—
441	C₂²×C₄	—	2	2	no	—
442	C₂²×C₄	—	2	2	no	—
443	C₂²×C₄	—	2	1	no	—
444	C₂²×C₄	—	2	1	no	—
445	C₂²×C₄	—	2	8	no	—
446	C₂²×C₄	—	2	1	no	—
447	C₂²×C₄	—	2	2	no	—
448	C₂²×C₄	—	4	4	no	—
449	C₂²×C₄	—	4	4	no	—
450	C₂²×C₄	C₂²×C₄×C₆₇	1	4	yes	—
451	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
452	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
453	C₂²×C₄	C₂×D₄×C₆₇	1	4	yes	—
454	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
455	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
456	C₂×C₈	—	2	1	no	—
457	C₂×C₈	—	2	1	no	—
458	C₂×C₈	—	2	1	no	—
459	C₂×C₈	—	2	2	no	—
460	C₂×C₈	—	2	2	no	—
461	C₂×C₈	—	2	1	no	—
462	C₂×C₈	—	2	1	no	—
463	C₂×C₈	—	2	2	no	—
464	C₂×C₈	—	4	8	no	—
465	C₂×C₈	C₂×D₄×C₆₇	1	2	yes	—
466	C₄⋊C₄	—	2	4	no	—
467	C₄⋊C₄	—	2	1	no	—
468	C₄⋊C₄	—	2	1	no	—
469	C₄⋊C₄	—	2	2	no	—
470	C₄⋊C₄	—	2	1	no	—
471	C₄⋊C₄	—	2	1	no	—
472	C₄⋊C₄	—	2	2	no	—
473	C₄⋊C₄	—	4	2	no	—
474	C₄⋊C₄	—	8	1	no	—
475	C₄⋊C₄	C₂²⋊C₄×C₆₇	1	4	yes	—
476	C₄⋊C₄	C₂²×C₄×C₆₇	1	2	yes	—
477	C₄⋊C₄	C₂×D₄×C₆₇	1	2	yes	—
478	C₂²⋊C₄	—	2	8	no	—
479	C₂²⋊C₄	—	4	4	no	—
480	C₂²⋊C₄	—	8	2	no	—
481	C₂²⋊C₄	C₂²⋊C₄×C₆₇	1	8	yes	—
482	C₄²	—	2	4	no	—
483	C₄²	—	4	4	no	—
484	C₄²	—	4	2	no	—
485	C₄²	C₂²×C₄×C₆₇	1	2	yes	—
486	C₄²	C₂×D₄×C₆₇	1	2	yes	—
487	C₂³	—	2	2	no	—
488	C₂³	—	2	2	no	—
489	C₂³	—	2	2	no	—
490	C₂³	—	2	2	no	—
491	C₂³	—	2	2	no	—
492	C₂³	—	2	2	no	—
493	C₂³	—	2	4	no	—
494	C₂³	—	4	2	no	—
495	C₂³	—	4	2	no	—
496	C₂³	—	4	2	no	—
497	C₂³	—	4	2	no	—
498	C₂³	—	4	2	no	—
499	C₂³	—	4	2	no	—
500	C₂³	C₂×C₂²⋊C₄×C₆₇	1	4	yes	—
501	C₂³	C₂²×D₄×C₆₇	1	1	yes	characteristic
502	Q₈	—	2	2	no	—
503	Q₈	—	2	2	no	—
504	Q₈	—	2	2	no	—
505	Q₈	C₄×D₄×C₆₇	1	4	yes	—
506	D₄	—	2	4	no	—
507	D₄	—	2	4	no	—
508	D₄	—	2	8	no	—
509	D₄	—	4	16	no	—
510	D₄	—	4	2	no	—
511	D₄	—	4	2	no	—
512	D₄	—	4	8	no	—
513	D₄	—	8	4	no	—
514	D₄	—	8	8	no	—
515	C₂×C₄	—	2	2	no	—
516	C₂×C₄	—	2	2	no	—
517	C₂×C₄	—	2	2	no	—
518	C₂×C₄	—	2	2	no	—
519	C₂×C₄	—	2	2	no	—
520	C₂×C₄	—	2	1	no	—
521	C₂×C₄	—	2	1	no	—
522	C₂×C₄	—	2	4	no	—
523	C₂×C₄	—	2	4	no	—
524	C₂×C₄	—	2	4	no	—
525	C₂×C₄	—	2	4	no	—
526	C₂×C₄	—	2	2	no	—
527	C₂×C₄	—	2	2	no	—
528	C₂×C₄	—	4	16	no	—
529	C₂×C₄	—	4	1	no	—
530	C₂×C₄	—	4	1	no	—
531	C₂×C₄	—	4	1	no	—
532	C₂×C₄	—	4	1	no	—
533	C₂×C₄	—	4	2	no	—
534	C₂×C₄	—	4	1	no	—
535	C₂×C₄	—	4	1	no	—
536	C₂×C₄	—	4	2	no	—
537	C₂×C₄	—	4	1	no	—
538	C₂×C₄	—	4	2	no	—
539	C₂×C₄	C₂×C₂²⋊C₄×C₆₇	1	2	yes	—
540	C₂×C₄	C₂×C₂²⋊C₄×C₆₇	1	4	yes	—
541	C₂×C₄	C₂×C₂²⋊C₄×C₆₇	1	2	yes	—
542	C₂×C₄	#135×C₆₇	1	2	yes	—
543	C₂×C₄	#135×C₆₇	1	2	yes	—
544	C₂×C₄	C₂³×C₄×C₆₇	1	1	yes	characteristic
545	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
546	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
547	C₂×C₄	C₂×#41×C₆₇	1	1	yes	characteristic
548	C₂×C₄	H(5,𝔽₂)×C₆₇	1	1	yes	characteristic
549	C₂×C₄	H(5,𝔽₂)×C₆₇	1	1	yes	characteristic
550	C₈	—	2	2	no	—
551	C₈	—	4	2	no	—
552	C₈	—	4	1	no	—
553	C₂²	—	2	4	no	—
554	C₂²	—	2	4	no	—
555	C₂²	—	2	4	no	—
556	C₂²	—	4	2	no	—
557	C₂²	—	4	2	no	—
558	C₂²	—	4	2	no	—
559	C₂²	—	4	2	no	—
560	C₂²	—	4	4	no	—
561	C₂²	—	4	4	no	—
562	C₂²	—	4	4	no	—
563	C₂²	—	8	2	no	—
564	C₂²	—	8	2	no	—
565	C₂²	—	8	2	no	—
566	C₂²	—	8	2	no	—
567	C₂²	C₂²×C₂²⋊C₄×C₆₇	1	1	yes	characteristic
568	C₂²	D₄²×C₆₇	1	1	yes	characteristic
569	C₂²	#546×C₆₇	1	1	yes	characteristic
570	C₂²	C₂×#135×C₆₇	1	1	yes	characteristic
571	C₂²	C₂×#135×C₆₇	1	1	yes	characteristic
572	C₂²	#576×C₆₇	1	1	yes	characteristic
573	C₂²	#577×C₆₇	1	1	yes	characteristic
574	C₄	—	2	4	no	—
575	C₄	—	2	4	no	—
576	C₄	—	2	2	no	—
577	C₄	—	4	2	no	—
578	C₄	—	4	2	no	—
579	C₄	—	8	2	no	—
580	C₄	—	8	1	no	—
581	C₄	#416×C₆₇	1	4	yes	complemented
582	C₄	C₂×C₄×D₄×C₆₇	1	1	yes	characteristic
583	C₄	#518×C₆₇	1	1	yes	characteristic
584	C₄	D₄²×C₆₇	1	1	yes	characteristic
585	C₄	#546×C₆₇	1	1	yes	characteristic
586	C₂	—	4	4	no	—
587	C₂	—	8	4	no	—
588	C₂	D₄×C₂²⋊C₄×C₆₇	1	1	yes	characteristic
589	C₂	C₂×#416×C₆₇	1	1	yes	characteristic
590	C₂	C₆₇×#2897	1	1	yes	characteristic
591	C₂	C₆₇×#3284	1	1	yes	characteristic
592	C₂	C₆₇×#3287	1	1	yes	characteristic
593	C₂	C₆₇×#3304	1	1	yes	characteristic
594	C₂	C₆₇×#3305	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₆₇	C₂²×D₄
commutator	C₂×C₄	C₂³×C₄×C₆₇
Frattini, cosocle	C₂²×C₄	C₂⁴×C₆₇
prime radical	C₂×#550×C₆₇	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₆₇ ◁ C₂×C₄×D₄×C₆₇ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×C₈×D₄×C₆₇	C₂	semidirect
3. C₂×#416×C₆₇	C₂	semidirect
4. C₆₇×#2354	C₂	semidirect
5. C₆₇×#1982	C₂	semidirect
6. C₆₇×#1957	C₂	semidirect
7. C₆₇×#1893	C₂	semidirect
8. C₆₇×#1875	C₂	semidirect
9. C₆₇×#1803	C₂	semidirect
10. C₆₇×#1788	C₂	semidirect
11. C₄×#103×C₆₇	C₂	semidirect
15. #550×C₆₇	C₄	semidirect
98. #103×C₆₇	D₄	semidirect
283. C₄×C₆₇	#416	semidirect
288. #22418	C₆₇	direct (N × H)
299. C₂×C₈×D₄	C₂×C₆₇	semidirect
300. C₂×#416	C₂×C₆₇	semidirect
301. #2354	C₂×C₆₇	semidirect
302. #1982	C₂×C₆₇	semidirect
303. #1957	C₂×C₆₇	semidirect
304. #1893	C₂×C₆₇	semidirect
305. #1875	C₂×C₆₇	semidirect
306. #1803	C₂×C₆₇	semidirect
307. #1788	C₂×C₆₇	semidirect
308. C₄×#103	C₂×C₆₇	semidirect
309. C₆₇	#22418	direct (N × H)
313. #550	C₄×C₆₇	semidirect
396. #103	D₄×C₆₇	semidirect
581. C₄	#416×C₆₇	semidirect

Maximal subgroups

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

subgroup	index
C₂×#550×C₆₇	2
C₂×C₈×D₄×C₆₇	2
C₂×#416×C₆₇	2
C₆₇×#2354	2
C₆₇×#1982	2
C₆₇×#1957	2
C₆₇×#1893	2
C₆₇×#1875	2
C₆₇×#1803	2
C₆₇×#1788	2
C₄×#103×C₆₇	2
#22418	67

Subgroups — C67×#22418 #30701669

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#22418 #30701669