Subgroups — #30853861

Order 17152.

1010 conjugacy classes of subgroups.

The subgroup lattice has 464 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#108×D₆₇	C₂	1	1	yes	maximal, characteristic
2	#427×D₆₇	C₂	1	1	yes	maximal, characteristic
3	#22568625	C₂	1	2	yes	maximal
4	C₂×#16863350	C₂	1	1	yes	maximal, characteristic
5	#22560094	C₂	1	2	yes	maximal, complemented
6	#22560092	C₂	1	2	yes	maximal, complemented
7	#22560091	C₂	1	1	yes	maximal, characteristic, complemented
8	#22560090	C₂	1	1	yes	maximal, characteristic, complemented
9	#22560088	C₂	1	1	yes	maximal, characteristic, complemented
10	#22559917	C₂	1	2	yes	maximal
11	C₆₇×#1772	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂²×#11761611	C₂²	1	1	yes	characteristic
13	C₂×M₁₆×D₆₇	C₂²	1	2	yes	—
14	C₂²×C₈×D₆₇	C₂²	1	1	yes	characteristic
15	C₂×#14524000	C₂²	1	1	yes	characteristic
16	#117×D₆₇	C₂²	1	2	yes	—
17	C₂×#14523979	C₂²	1	1	yes	characteristic
18	C₂×#14523978	C₂²	1	2	yes	—
19	#108×D₆₇	C₄	1	8	yes	complemented
20	#16863434	C₂²	1	1	yes	characteristic
21	#16863433	C₄	1	2	yes	complemented
22	#16863432	C₂²	1	2	yes	—
23	#16863431	C₄	1	2	yes	complemented
24	#107×D₆₇	C₄	1	2	yes	complemented
25	#106×D₆₇	C₄	1	2	yes	complemented
26	#16863354	C₂²	1	2	yes	—
27	#16863353	—	2	2	no	—
28	#16863352	C₂²	1	2	yes	—
29	#16863351	C₂²	1	2	yes	—
30	#16863350	—	2	2	no	—
31	#16863350	C₄	1	8	yes	complemented
32	#16863348	—	2	2	no	—
33	C₄×#11761611	—	2	2	no	—
34	C₂×#14523954	C₂²	1	1	yes	characteristic
35	C₂×#14523953	C₂²	1	1	yes	characteristic
36	C₂×C₈×Dic₆₇	C₂²	1	1	yes	characteristic
37	#16863157	C₂²	1	2	yes	—
38	#16863156	C₂²	1	2	yes	—
39	#16863155	C₂²	1	2	yes	—
40	C₂×#108×C₆₇	C₂²	1	1	yes	characteristic
41	#427×C₆₇	C₂²	1	1	yes	characteristic
42	#405×C₆₇	C₂²	1	2	yes	—
43	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
44	#339×C₆₇	C₂²	1	2	yes	—
45	C₂²×C₄×D₆₇	C₂³	1	1	yes	characteristic
46	C₂×#11761612	C₂³	1	2	yes	—
47	C₂×#11761611	—	2	2	no	—
48	C₂×#11761611	C₂×C₄	1	4	yes	—
49	C₂×#11761611	D₄	1	1	yes	characteristic
50	C₂×#11761611	D₄	1	1	yes	characteristic
51	C₂×#11761611	D₄	1	1	yes	characteristic
52	C₂×#11761611	D₄	1	1	yes	characteristic
53	C₂×#11761611	Q₈	1	4	yes	—
54	M₁₆×D₆₇	—	2	4	no	—
55	M₁₆×D₆₇	C₂×C₄	1	8	yes	—
56	C₂²×C₆₇⋊C₈	C₂³	1	1	yes	characteristic
57	C₂²×C₆₇⋊C₈	C₂³	1	1	yes	characteristic
58	C₂×C₈×D₆₇	—	2	2	no	—
59	C₂×C₈×D₆₇	C₂×C₄	1	4	yes	—
60	C₂×C₈×D₆₇	C₂×C₄	1	1	yes	characteristic
61	C₂×C₈×D₆₇	C₂×C₄	1	1	yes	characteristic
62	C₂×C₈×D₆₇	C₂×C₄	1	1	yes	characteristic
63	C₂×C₈×D₆₇	C₂×C₄	1	1	yes	characteristic
64	C₂×C₈×D₆₇	C₂×C₄	1	4	yes	—
65	#14524002	C₂³	1	2	yes	—
66	#14524001	C₂×C₄	1	4	yes	—
67	#14524000	—	2	2	no	—
68	#14524000	—	2	2	no	—
69	#14524000	C₂×C₄	1	4	yes	—
70	#14523999	—	2	2	no	—
71	C₂×Dic₆₇⋊C₄	—	2	1	no	—
72	C₄⋊C₄×D₆₇	C₂×C₄	1	4	yes	—
73	C₂×(C₂×D₆₇)⋊C₄	—	2	1	no	—
74	C₂²⋊C₄×D₆₇	—	2	2	no	—
75	C₂×C₄×Dic₆₇	C₂³	1	1	yes	characteristic
76	C₄²×D₆₇	—	2	2	no	—
77	#14523979	C₂×C₄	1	4	yes	complemented
78	#14523978	C₂×C₄	1	8	yes	complemented
79	#14523977	C₂×C₄	1	2	yes	—
80	#14523976	C₂×C₄	1	4	yes	—
81	#14523975	C₂×C₄	1	2	yes	—
82	#14523974	C₂×C₄	1	4	yes	—
83	#14523954	—	2	2	no	—
84	#14523954	C₂×C₄	1	4	yes	complemented
85	#14523953	—	2	2	no	—
86	#14523953	—	2	2	no	—
87	#14523953	C₂×C₄	1	4	yes	complemented
88	C₄×C₆₇⋊C₈	—	2	2	no	—
89	C₈×Dic₆₇	—	2	2	no	—
90	C₈×Dic₆₇	—	2	2	no	—
91	C₈×Dic₆₇	C₂×C₄	1	4	yes	complemented
92	C₂×M₁₆×C₆₇	C₂³	1	2	yes	—
93	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
94	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
95	#117×C₆₇	C₂³	1	2	yes	—
96	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
97	#108×C₆₇	C₂×C₄	1	4	yes	complemented
98	#107×C₆₇	C₂×C₄	1	2	yes	—
99	#106×C₆₇	C₂×C₄	1	2	yes	—
100	#97×C₆₇	—	2	2	no	—
101	C₄×C₈×C₆₇	—	2	2	no	—
102	C₄×C₈×C₆₇	C₂×C₄	1	4	yes	complemented
103	C₂³×D₆₇	#41	1	1	yes	characteristic
104	C₂²×Dic₆₇	—	2	1	no	—
105	C₂²×Dic₆₇	#41	1	1	yes	characteristic
106	C₂×C₄×D₆₇	—	2	4	no	—
107	C₂×C₄×D₆₇	C₂²×C₄	1	2	yes	—
108	C₂×C₄×D₆₇	C₂²×C₄	1	2	yes	—
109	C₂×C₄×D₆₇	C₂²×C₄	1	1	yes	characteristic
110	C₂×C₄×D₆₇	C₂²×C₄	1	1	yes	characteristic
111	C₂×C₄×D₆₇	C₂×D₄	1	1	yes	characteristic
112	C₂×C₄×D₆₇	C₂×D₄	1	1	yes	characteristic
113	C₂×C₄×D₆₇	C₂×Q₈	1	2	yes	—
114	C₂×C₄×D₆₇	#41	1	2	yes	—
115	#11761612	—	2	2	no	—
116	#11761612	C₂²×C₄	1	4	yes	—
117	#11761611	—	2	8	no	—
118	#11761611	C₄⋊C₄	1	4	yes	—
119	#11761611	C₄⋊C₄	1	4	yes	—
120	#11761611	C₄⋊C₄	1	4	yes	—
121	#11761611	C₄⋊C₄	1	4	yes	—
122	C₂×C₆₇⋊C₈	—	2	2	no	—
123	C₂×C₆₇⋊C₈	C₂²×C₄	1	2	yes	—
124	C₂×C₆₇⋊C₈	C₂²×C₄	1	1	yes	characteristic
125	C₂×C₆₇⋊C₈	C₂²×C₄	1	1	yes	characteristic
126	C₂×C₆₇⋊C₈	C₂²×C₄	1	2	yes	—
127	C₂×C₆₇⋊C₈	C₂²×C₄	1	2	yes	—
128	C₂×C₆₇⋊C₈	C₂×D₄	1	1	yes	characteristic
129	C₂×C₆₇⋊C₈	C₂×D₄	1	1	yes	characteristic
130	C₂×C₆₇⋊C₈	C₂×Q₈	1	2	yes	—
131	C₈×D₆₇	—	2	8	no	—
132	C₈×D₆₇	C₄²	1	4	yes	—
133	C₈×D₆₇	C₄²	1	4	yes	—
134	C₈×D₆₇	C₄²	1	4	yes	—
135	C₈×D₆₇	C₄²	1	4	yes	—
136	C₄⋊Dic₆₇	C₂²×C₄	1	4	yes	—
137	Dic₆₇⋊C₄	—	2	4	no	—
138	Dic₆₇⋊C₄	—	2	4	no	—
139	C₂²⋊Dic₆₇	—	2	2	no	—
140	(C₂×D₆₇)⋊C₄	—	2	4	no	—
141	(C₂×D₆₇)⋊C₄	—	2	4	no	—
142	C₄×Dic₆₇	—	2	2	no	—
143	C₄×Dic₆₇	—	2	1	no	—
144	C₄×Dic₆₇	—	2	4	no	—
145	C₄×Dic₆₇	C₂²×C₄	1	2	yes	—
146	C₂²×C₄×C₆₇	—	2	1	no	—
147	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
148	M₁₆×C₆₇	—	2	2	no	—
149	M₁₆×C₆₇	C₂²×C₄	1	4	yes	—
150	C₂×C₈×C₆₇	—	2	2	no	—
151	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
152	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
153	C₂×C₈×C₆₇	C₂²×C₄	1	2	yes	—
154	C₂×C₈×C₆₇	C₂²×C₄	1	2	yes	—
155	C₂×C₈×C₆₇	C₂²×C₄	1	2	yes	—
156	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
157	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
158	C₂×C₈×C₆₇	C₂×Q₈	1	2	yes	—
159	C₄⋊C₄×C₆₇	C₂²×C₄	1	4	yes	—
160	C₂²⋊C₄×C₆₇	—	2	2	no	—
161	C₄²×C₆₇	—	2	2	no	—
162	C₄²×C₆₇	—	2	1	no	—
163	C₄²×C₆₇	C₂²×C₄	1	2	yes	—
164	C₂³×C₆₇	C₂×#41	1	1	yes	characteristic
165	C₂²×D₆₇	—	4	2	no	—
166	C₂²×D₆₇	#117	1	2	yes	—
167	C₂²×D₆₇	C₄×D₄	1	1	yes	characteristic
168	C₂²×D₆₇	C₄×D₄	1	1	yes	characteristic
169	C₂²×D₆₇	C₄×Q₈	1	2	yes	—
170	C₂×C₄×C₆₇	—	2	2	no	—
171	C₂×C₄×C₆₇	—	2	4	no	—
172	C₂×C₄×C₆₇	—	4	2	no	—
173	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
174	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
175	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
176	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
177	C₂×C₄×C₆₇	C₂²×Q₈	1	1	yes	characteristic
178	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
179	C₂×Dic₆₇	—	2	2	no	—
180	C₂×Dic₆₇	—	2	4	no	—
181	C₂×Dic₆₇	—	4	2	no	—
182	C₂×Dic₆₇	#117	1	2	yes	—
183	C₂×Dic₆₇	C₄×D₄	1	1	yes	characteristic
184	C₂×Dic₆₇	C₄×D₄	1	1	yes	characteristic
185	C₂×Dic₆₇	C₄×Q₈	1	2	yes	—
186	C₄×D₆₇	—	8	4	no	—
187	C₄×D₆₇	C₂×C₄²	1	2	yes	—
188	C₄×D₆₇	C₂×C₄²	1	2	yes	—
189	C₄×D₆₇	C₂×C₄⋊C₄	1	2	yes	—
190	C₄×D₆₇	C₂×C₄⋊C₄	1	2	yes	—
191	C₄×D₆₇	C₄×D₄	1	2	yes	—
192	C₄×D₆₇	C₄×D₄	1	2	yes	—
193	C₄×D₆₇	C₄×Q₈	1	2	yes	—
194	C₄×D₆₇	C₄×Q₈	1	2	yes	—
195	C₈×C₆₇	—	2	4	no	—
196	C₈×C₆₇	C₂×C₄²	1	2	yes	—
197	C₈×C₆₇	C₂×C₄²	1	2	yes	—
198	C₈×C₆₇	C₂×C₄⋊C₄	1	2	yes	—
199	C₈×C₆₇	C₂×C₄⋊C₄	1	2	yes	—
200	C₆₇⋊C₈	—	2	4	no	—
201	C₆₇⋊C₈	C₂×C₄²	1	2	yes	—
202	C₆₇⋊C₈	C₂×C₄²	1	2	yes	—
203	C₆₇⋊C₈	C₂×C₄⋊C₄	1	2	yes	—
204	C₆₇⋊C₈	C₂×C₄⋊C₄	1	2	yes	—
205	C₂²×C₆₇	—	2	2	no	—
206	C₂²×C₆₇	C₂×#117	1	1	yes	characteristic
207	C₂²×C₆₇	C₂×C₄×D₄	1	1	yes	characteristic
208	C₂²×C₆₇	C₂×C₄×Q₈	1	1	yes	characteristic
209	C₂×D₆₇	—	2	2	no	—
210	C₂×D₆₇	—	2	2	no	—
211	C₂×D₆₇	—	4	2	no	—
212	C₂×D₆₇	—	4	2	no	—
213	C₂×D₆₇	C₄×C₄⋊C₄	1	2	yes	—
214	C₂×D₆₇	C₄×C₄⋊C₄	1	2	yes	—
215	C₄×C₆₇	—	4	2	no	—
216	C₄×C₆₇	—	4	4	no	—
217	C₄×C₆₇	C₂²×C₄²	1	1	yes	characteristic
218	C₄×C₆₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
219	C₄×C₆₇	C₂×C₄×D₄	1	1	yes	characteristic
220	C₄×C₆₇	C₂×C₄×Q₈	1	1	yes	characteristic
221	Dic₆₇	—	4	2	no	—
222	Dic₆₇	—	4	4	no	—
223	Dic₆₇	C₄×C₄⋊C₄	1	2	yes	—
224	Dic₆₇	C₄×C₄⋊C₄	1	2	yes	—
225	#20446	—	67	1	no	maximal, Sylow
226	C₂×C₆₇	—	2	1	no	—
227	C₂×C₆₇	—	2	1	no	—
228	C₂×C₆₇	C₂×C₄×C₄⋊C₄	1	1	yes	characteristic
229	C₂×C₆₇	#2957	1	2	yes	—
230	D₆₇	—	2	2	no	—
231	D₆₇	—	2	2	no	—
232	C₂²×#108	—	67	1	no	—
233	C₂×#427	—	67	1	no	—
234	#2873	—	67	2	no	—
235	C₂×#405	—	67	1	no	—
236	#1773	—	67	2	no	—
237	#1773	—	67	2	no	—
238	#1772	—	67	1	no	—
239	#1772	—	67	1	no	—
240	#1772	—	67	1	no	—
241	#1772	D₆₇	1	1	yes	characteristic, complemented
242	#1733	—	67	2	no	—
243	C₆₇	#20446	1	1	yes	characteristic, complemented, Sylow
244	C₂²×M₁₆	—	67	1	no	—
245	C₂²×M₁₆	—	67	2	no	—
246	C₂³×C₈	—	67	1	no	—
247	C₂×#117	—	67	1	no	—
248	C₂×#117	—	67	2	no	—
249	C₂×#108	—	67	2	no	—
250	C₂×#108	—	67	1	no	—
251	C₂×#108	—	67	8	no	—
252	C₂×#108	C₂×D₆₇	1	1	yes	characteristic
253	#427	—	67	1	no	—
254	#427	—	67	2	no	—
255	#427	—	67	2	no	—
256	#427	—	67	2	no	—
257	#427	C₂×D₆₇	1	1	yes	characteristic
258	C₂×#107	—	67	2	no	—
259	C₂×#106	—	67	2	no	—
260	#405	—	67	8	no	—
261	#405	—	67	2	no	—
262	#405	—	67	2	no	—
263	#405	—	67	2	no	—
264	#405	—	134	2	no	—
265	#405	—	134	2	no	—
266	#405	C₂×D₆₇	1	2	yes	—
267	C₄×M₁₆	—	134	2	no	—
268	C₄×M₁₆	—	134	2	no	—
269	C₂×#97	—	67	1	no	—
270	C₂×#97	—	67	1	no	—
271	C₂×C₄×C₈	—	67	1	no	—
272	C₂×C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
273	#339	—	67	2	no	—
274	#339	—	67	2	no	—
275	#339	—	67	2	no	—
276	#339	C₂×D₆₇	1	2	yes	—
277	C₂³×C₄	—	67	1	no	—
278	C₂×M₁₆	—	67	4	no	—
279	C₂×M₁₆	—	67	1	no	—
280	C₂×M₁₆	—	67	1	no	—
281	C₂×M₁₆	—	67	1	no	—
282	C₂×M₁₆	—	67	1	no	—
283	C₂×M₁₆	—	67	4	no	—
284	C₂×M₁₆	—	67	2	no	—
285	C₂×M₁₆	—	67	8	no	—
286	C₂×M₁₆	—	134	2	no	—
287	C₂×M₁₆	—	134	4	no	—
288	C₂×M₁₆	C₂²×D₆₇	1	2	yes	—
289	C₂²×C₈	—	67	1	no	—
290	C₂²×C₈	—	67	1	no	—
291	C₂²×C₈	—	67	4	no	—
292	C₂²×C₈	—	67	1	no	—
293	C₂²×C₈	—	67	1	no	—
294	C₂²×C₈	—	67	1	no	—
295	C₂²×C₈	—	67	1	no	—
296	C₂²×C₈	—	67	4	no	—
297	C₂²×C₈	—	134	2	no	—
298	C₂²×C₈	C₂²×D₆₇	1	1	yes	characteristic
299	C₂²×C₈	C₂²×D₆₇	1	1	yes	characteristic
300	#117	—	67	4	no	—
301	#117	—	67	2	no	—
302	#117	—	67	4	no	—
303	#117	—	134	2	no	—
304	#117	—	134	2	no	—
305	#117	—	134	2	no	—
306	#117	C₂²×D₆₇	1	2	yes	—
307	C₂×C₄⋊C₄	—	67	4	no	—
308	C₂×C₄⋊C₄	—	134	1	no	—
309	C₂×C₂²⋊C₄	—	134	2	no	—
310	C₂×C₂²⋊C₄	—	134	1	no	—
311	C₂×C₄²	—	67	1	no	—
312	C₂×C₄²	—	134	2	no	—
313	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
314	#108	—	67	8	no	—
315	#108	—	67	4	no	—
316	#108	C₄×D₆₇	1	4	yes	complemented
317	#107	—	67	2	no	—
318	#107	—	67	4	no	—
319	#107	C₄×D₆₇	1	2	yes	—
320	#106	—	67	2	no	—
321	#106	—	67	4	no	—
322	#106	C₄×D₆₇	1	2	yes	—
323	#97	—	2	2	no	—
324	#97	—	67	4	no	—
325	#97	—	67	4	no	—
326	#97	—	134	2	no	—
327	#97	—	134	2	no	—
328	#97	—	134	2	no	—
329	C₄×C₈	—	2	2	no	—
330	C₄×C₈	—	67	4	no	—
331	C₄×C₈	—	134	2	no	—
332	C₄×C₈	—	134	2	no	—
333	C₄×C₈	—	134	2	no	—
334	C₄×C₈	C₄×D₆₇	1	4	yes	complemented
335	C₂⁴	—	67	1	no	—
336	C₂²×C₄	—	2	1	no	—
337	C₂²×C₄	—	67	1	no	—
338	C₂²×C₄	—	67	1	no	—
339	C₂²×C₄	—	67	1	no	—
340	C₂²×C₄	—	67	2	no	—
341	C₂²×C₄	—	67	2	no	—
342	C₂²×C₄	—	67	1	no	—
343	C₂²×C₄	—	67	1	no	—
344	C₂²×C₄	—	67	2	no	—
345	C₂²×C₄	—	67	2	no	—
346	C₂²×C₄	—	134	4	no	—
347	C₂²×C₄	—	134	1	no	—
348	C₂²×C₄	C₂³×D₆₇	1	1	yes	characteristic
349	M₁₆	—	2	2	no	—
350	M₁₆	—	67	4	no	—
351	M₁₆	—	67	4	no	—
352	M₁₆	—	67	4	no	—
353	M₁₆	—	67	4	no	—
354	M₁₆	—	67	4	no	—
355	M₁₆	—	134	8	no	—
356	M₁₆	—	134	2	no	—
357	M₁₆	C₂×C₄×D₆₇	1	4	yes	—
358	C₂×C₈	—	2	2	no	—
359	C₂×C₈	—	67	2	no	—
360	C₂×C₈	—	67	2	no	—
361	C₂×C₈	—	67	1	no	—
362	C₂×C₈	—	67	1	no	—
363	C₂×C₈	—	67	2	no	—
364	C₂×C₈	—	67	1	no	—
365	C₂×C₈	—	67	1	no	—
366	C₂×C₈	—	67	2	no	—
367	C₂×C₈	—	67	4	no	—
368	C₂×C₈	—	67	4	no	—
369	C₂×C₈	—	67	4	no	—
370	C₂×C₈	—	67	4	no	—
371	C₂×C₈	—	134	8	no	—
372	C₂×C₈	—	134	2	no	—
373	C₂×C₈	C₂×C₄×D₆₇	1	2	yes	—
374	C₂×C₈	C₂×C₄×D₆₇	1	1	yes	characteristic
375	C₂×C₈	C₂×C₄×D₆₇	1	1	yes	characteristic
376	C₂×C₈	C₂×C₄×D₆₇	1	2	yes	—
377	C₂×C₈	C₂×C₄×D₆₇	1	2	yes	—
378	C₂×C₈	D₄×D₆₇	1	1	yes	characteristic
379	C₂×C₈	D₄×D₆₇	1	1	yes	characteristic
380	C₂×C₈	Q₈×D₆₇	1	2	yes	—
381	C₄⋊C₄	—	67	4	no	—
382	C₄⋊C₄	—	134	4	no	—
383	C₄⋊C₄	—	134	4	no	—
384	C₄⋊C₄	C₂×C₄×D₆₇	1	4	yes	—
385	C₂²⋊C₄	—	2	2	no	—
386	C₂²⋊C₄	—	134	4	no	—
387	C₂²⋊C₄	—	134	4	no	—
388	C₂²⋊C₄	—	134	2	no	—
389	C₄²	—	2	2	no	—
390	C₄²	—	2	1	no	—
391	C₄²	—	67	2	no	—
392	C₄²	—	134	4	no	—
393	C₄²	—	134	2	no	—
394	C₄²	—	134	1	no	—
395	C₄²	C₂×C₄×D₆₇	1	2	yes	—
396	C₂³	—	268	2	no	—
397	C₂³	—	67	2	no	—
398	C₂³	—	67	2	no	—
399	C₂³	—	67	1	no	—
400	C₂³	—	67	1	no	—
401	C₂³	#41×D₆₇	1	1	yes	characteristic
402	C₂×C₄	—	2	2	no	—
403	C₂×C₄	—	2	4	no	—
404	C₂×C₄	—	4	2	no	—
405	C₂×C₄	—	268	2	no	—
406	C₂×C₄	—	536	4	no	—
407	C₂×C₄	—	67	2	no	—
408	C₂×C₄	—	67	1	no	—
409	C₂×C₄	—	67	1	no	—
410	C₂×C₄	—	67	2	no	—
411	C₂×C₄	—	67	2	no	—
412	C₂×C₄	—	67	2	no	—
413	C₂×C₄	—	67	2	no	—
414	C₂×C₄	—	67	2	no	—
415	C₂×C₄	—	67	2	no	—
416	C₂×C₄	—	67	2	no	—
417	C₂×C₄	—	67	2	no	—
418	C₂×C₄	—	67	2	no	—
419	C₂×C₄	—	134	2	no	—
420	C₂×C₄	—	134	4	no	—
421	C₂×C₄	C₂²×C₄×D₆₇	1	1	yes	characteristic
422	C₂×C₄	C₂²×C₄×D₆₇	1	1	yes	characteristic
423	C₂×C₄	C₂²×C₄×D₆₇	1	1	yes	characteristic
424	C₂×C₄	C₂×D₄×D₆₇	1	1	yes	characteristic
425	C₂×C₄	C₂×Q₈×D₆₇	1	1	yes	characteristic
426	C₂×C₄	#41×D₆₇	1	1	yes	characteristic
427	C₈	—	2	4	no	—
428	C₈	—	67	2	no	—
429	C₈	—	67	2	no	—
430	C₈	—	67	2	no	—
431	C₈	—	67	2	no	—
432	C₈	—	134	4	no	—
433	C₈	C₄²×D₆₇	1	2	yes	—
434	C₈	C₄²×D₆₇	1	2	yes	—
435	C₈	C₄⋊C₄×D₆₇	1	2	yes	—
436	C₈	C₄⋊C₄×D₆₇	1	2	yes	—
437	C₂²	—	2	2	no	—
438	C₂²	—	268	2	no	—
439	C₂²	—	268	2	no	—
440	C₂²	—	67	2	no	—
441	C₂²	—	67	2	no	—
442	C₂²	—	134	2	no	—
443	C₂²	—	134	2	no	—
444	C₂²	#117×D₆₇	1	1	yes	characteristic
445	C₂²	C₄×D₄×D₆₇	1	1	yes	characteristic
446	C₂²	C₄×Q₈×D₆₇	1	1	yes	characteristic
447	C₄	—	4	2	no	—
448	C₄	—	4	4	no	—
449	C₄	—	268	2	no	—
450	C₄	—	268	4	no	—
451	C₄	—	67	2	no	—
452	C₄	—	67	2	no	—
453	C₄	C₂×C₄²×D₆₇	1	1	yes	characteristic
454	C₄	C₂×C₄⋊C₄×D₆₇	1	1	yes	characteristic
455	C₄	C₄×D₄×D₆₇	1	1	yes	characteristic
456	C₄	C₄×Q₈×D₆₇	1	1	yes	characteristic
457	C₂	—	2	1	no	—
458	C₂	—	2	1	no	—
459	C₂	—	134	2	no	—
460	C₂	—	134	2	no	—
461	C₂	C₄×C₄⋊C₄×D₆₇	1	1	yes	characteristic
462	C₂	#22569121	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₂×D₄×D₆₇
commutator	C₄×C₆₇	C₂²×C₄²
Frattini	C₂²×C₄	C₂³×D₆₇
Fitting	C₆₇×#1772	C₂
socle	C₂²×C₆₇	C₂×C₄×D₄
prime radical	C₂³×D₆₇	#41
cosocle	C₂²×C₄×C₆₇	C₂⁴
hypercenter	#1772	D₆₇
hypocenter	C₆₇	#20446
LCS	G ▷ C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₈ ◁ #1772

Complements

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

N	H ≅ G/N	type
5. #22560094	C₂	semidirect
6. #22560092	C₂	semidirect
7. #22560091	C₂	semidirect
8. #22560090	C₂	semidirect
9. #22560088	C₂	semidirect
11. C₆₇×#1772	C₂	semidirect
19. #108×D₆₇	C₄	semidirect
21. #16863433	C₄	semidirect
23. #16863431	C₄	semidirect
24. #107×D₆₇	C₄	semidirect
25. #106×D₆₇	C₄	semidirect
31. #16863350	C₄	semidirect
77. #14523979	C₂×C₄	semidirect
78. #14523978	C₂×C₄	semidirect
84. #14523954	C₂×C₄	semidirect
87. #14523953	C₂×C₄	semidirect
91. C₈×Dic₆₇	C₂×C₄	semidirect
97. #108×C₆₇	C₂×C₄	semidirect
102. C₄×C₈×C₆₇	C₂×C₄	semidirect
241. #1772	D₆₇	semidirect
243. C₆₇	#20446	semidirect
316. #108	C₄×D₆₇	semidirect
334. C₄×C₈	C₄×D₆₇	semidirect

Maximal subgroups

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

subgroup	index
C₂×#108×D₆₇	2
#427×D₆₇	2
#22568625	2
C₂×#16863350	2
#22560094	2
#22560092	2
#22560091	2
#22560090	2
#22560088	2
#22559917	2
C₆₇×#1772	2
#20446	67