Subgroups — #30868132

Order 17152.

648 conjugacy classes of subgroups.

The subgroup lattice has 490 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22569991	C₂	1	1	yes	maximal, characteristic
2	#22568851	C₂	1	1	yes	maximal, characteristic
3	#22561154	C₂	1	1	yes	maximal, characteristic, complemented
4	#22560982	C₂	1	1	yes	maximal, characteristic, complemented
5	#22560954	C₂	1	2	yes	maximal, complemented
6	#22560912	C₂	1	2	yes	maximal
7	#22560857	C₂	1	2	yes	maximal
8	#22560808	C₂	1	2	yes	maximal, complemented
9	#22560226	C₂	1	1	yes	maximal, characteristic
10	C₈×C₄⋊Dic₆₇	C₂	1	1	yes	maximal, characteristic, complemented
11	C₆₇×#2960	C₂	1	1	yes	maximal, characteristic, complemented
12	C₄×#11761628	C₂²	1	1	yes	characteristic
13	#16863700	C₄	1	2	yes	complemented
14	#16863693	C₄	1	2	yes	complemented
15	#16863688	C₄	1	1	yes	characteristic, complemented
16	#16863683	C₄	1	1	yes	characteristic, complemented
17	#16863678	C₄	1	1	yes	characteristic, complemented
18	#16863673	C₄	1	1	yes	characteristic, complemented
19	#16863455	—	2	1	no	—
20	#16863455	—	2	1	no	—
21	#16863447	—	2	1	no	—
22	#16863447	—	2	1	no	—
23	C₂×#14523978	C₂²	1	2	yes	—
24	#16863432	C₂²	1	2	yes	—
25	#16863422	C₂²	1	1	yes	characteristic
26	#16863422	C₂²	1	1	yes	characteristic
27	C₂×C₄⋊(C₆₇⋊C₈)	C₂²	1	1	yes	characteristic
28	C₂×#14523969	C₂²	1	2	yes	—
29	#16863390	C₂²	1	2	yes	—
30	#16863362	C₂²	1	1	yes	characteristic
31	#16863362	C₂²	1	1	yes	characteristic
32	C₄×#11761612	C₂²	1	1	yes	characteristic
33	C₂×C₈×Dic₆₇	C₂²	1	2	yes	—
34	C₄×C₄⋊Dic₆₇	C₂²	1	1	yes	characteristic
35	#16863160	C₂²	1	2	yes	—
36	#16863157	C₂²	1	2	yes	—
37	#16863155	C₂²	1	2	yes	—
38	#16863155	C₂²	1	2	yes	—
39	#16863150	C₂²	1	2	yes	—
40	#533×C₆₇	C₂²	1	1	yes	characteristic
41	#446×C₆₇	—	2	1	no	—
42	#446×C₆₇	—	2	1	no	—
43	C₈×Q₈×C₆₇	—	2	1	no	—
44	C₈×Q₈×C₆₇	—	2	1	no	—
45	#424×C₆₇	C₂²	1	2	yes	—
46	#423×C₆₇	C₂²	1	1	yes	characteristic
47	#405×C₆₇	C₂²	1	2	yes	—
48	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
49	C₂×#11761628	C₂³	1	1	yes	characteristic
50	C₂×#11761612	C₂³	1	1	yes	characteristic
51	C₂×#11761612	C₂³	1	1	yes	characteristic
52	C₂²×C₆₇⋊C₈	C₂³	1	2	yes	—
53	C₄×Dic₁₃₄	C₂×C₄	1	1	yes	characteristic
54	C₄×Dic₁₃₄	C₂×C₄	1	1	yes	characteristic
55	C₄×D₂₆₈	C₂×C₄	1	1	yes	characteristic
56	C₄×D₂₆₈	C₂×C₄	1	1	yes	characteristic
57	C₄×C₂²⋊D₆₇	—	2	1	no	—
58	C₄×C₂²⋊D₆₇	—	2	1	no	—
59	#14524002	C₂³	1	1	yes	characteristic
60	#14524000	—	2	1	no	—
61	C₂×C₄⋊Dic₆₇	—	2	1	no	—
62	C₂×C₄×Dic₆₇	C₂³	1	2	yes	—
63	C₄²×D₆₇	—	2	1	no	—
64	#14523978	—	4	2	no	—
65	#14523976	C₂×C₄	1	2	yes	—
66	#14523976	C₂×C₄	1	2	yes	—
67	#14523974	C₂×C₄	1	2	yes	—
68	#14523974	C₂×C₄	1	2	yes	—
69	C₄⋊(C₆₇⋊C₈)	C₂×C₄	1	1	yes	characteristic
70	C₄⋊(C₆₇⋊C₈)	C₂×C₄	1	1	yes	characteristic
71	C₄⋊(C₆₇⋊C₈)	C₂×C₄	1	2	yes	—
72	Dic₆₇⋊C₈	—	2	1	no	—
73	Dic₆₇⋊C₈	—	2	1	no	—
74	Dic₆₇⋊C₈	—	2	1	no	—
75	Dic₆₇⋊C₈	—	2	1	no	—
76	#14523969	—	4	2	no	—
77	#14523966	C₂×C₄	1	2	yes	—
78	#14523966	C₂×C₄	1	2	yes	—
79	#14523963	C₂×C₄	1	2	yes	—
80	#14523963	C₂×C₄	1	2	yes	—
81	(C₂×D₆₇)⋊C₈	—	2	1	no	—
82	(C₂×D₆₇)⋊C₈	—	2	1	no	—
83	(C₂×D₆₇)⋊C₈	—	2	1	no	—
84	(C₂×D₆₇)⋊C₈	—	2	1	no	—
85	#14523954	—	2	1	no	—
86	C₄×C₆₇⋊C₈	—	2	1	no	—
87	C₈×Dic₆₇	—	4	2	no	—
88	C₆₇⋊#95	—	2	1	no	—
89	C₂×M₁₆×C₆₇	C₂³	1	2	yes	—
90	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
91	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
92	C₄⋊Q₈×C₆₇	C₂×C₄	1	1	yes	characteristic
93	C₄⋊Q₈×C₆₇	C₂×C₄	1	1	yes	characteristic
94	#125×C₆₇	C₂×C₄	1	2	yes	—
95	C₂²⋊Q₈×C₆₇	—	2	1	no	—
96	C₂²⋊Q₈×C₆₇	—	2	1	no	—
97	C₄×Q₈×C₆₇	—	2	1	no	—
98	C₄×Q₈×C₆₇	—	2	1	no	—
99	#117×C₆₇	C₂³	1	2	yes	—
100	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
101	C₄⋊C₈×C₆₇	—	2	2	no	—
102	C₄⋊C₈×C₆₇	—	2	2	no	—
103	C₄⋊C₈×C₆₇	—	2	1	no	—
104	C₄⋊C₈×C₆₇	—	2	1	no	—
105	#97×C₆₇	—	2	2	no	—
106	C₄×C₈×C₆₇	—	2	2	no	—
107	C₄×C₈×C₆₇	D₄	1	1	yes	characteristic
108	C₄×C₈×C₆₇	D₄	1	1	yes	characteristic
109	C₄×C₈×C₆₇	D₄	1	1	yes	characteristic
110	C₄×C₈×C₆₇	D₄	1	1	yes	characteristic
111	#11761628	—	2	1	no	—
112	#11761628	—	2	1	no	—
113	#11761628	—	2	1	no	—
114	#11761628	—	2	1	no	—
115	C₂×Dic₁₃₄	C₂²×C₄	1	1	yes	characteristic
116	C₂×D₂₆₈	C₂²×C₄	1	1	yes	characteristic
117	C₂×C₂²⋊D₆₇	—	2	1	no	—
118	C₂²×Dic₆₇	—	2	2	no	—
119	C₂×C₄×D₆₇	—	2	1	no	—
120	C₂×C₄×D₆₇	—	2	1	no	—
121	C₂×C₄×D₆₇	—	2	1	no	—
122	#11761612	—	2	1	no	—
123	#11761612	—	2	1	no	—
124	#11761612	—	2	1	no	—
125	#11761612	—	2	1	no	—
126	C₂×C₆₇⋊C₈	—	2	2	no	—
127	C₂×C₆₇⋊C₈	—	2	2	no	—
128	C₂×C₆₇⋊C₈	—	2	1	no	—
129	C₂×C₆₇⋊C₈	—	2	1	no	—
130	C₂×C₆₇⋊C₈	C₂²×C₄	1	2	yes	—
131	C₂×C₆₇⋊C₈	C₂²×C₄	1	2	yes	—
132	C₄⋊Dic₆₇	—	2	4	no	—
133	C₄⋊Dic₆₇	C₂²×C₄	1	1	yes	characteristic
134	C₄⋊Dic₆₇	C₂²×C₄	1	1	yes	characteristic
135	Dic₆₇⋊C₄	—	2	1	no	—
136	Dic₆₇⋊C₄	—	2	1	no	—
137	C₂²⋊Dic₆₇	—	2	1	no	—
138	(C₂×D₆₇)⋊C₄	—	2	1	no	—
139	(C₂×D₆₇)⋊C₄	—	2	1	no	—
140	C₄×Dic₆₇	—	2	1	no	—
141	C₄×Dic₆₇	—	2	1	no	—
142	C₄×Dic₆₇	—	2	1	no	—
143	C₄×Dic₆₇	—	2	2	no	—
144	C₄×Dic₆₇	—	2	2	no	—
145	C₂×Q₈×C₆₇	—	2	1	no	—
146	C₂×Q₈×C₆₇	—	2	1	no	—
147	C₂²×C₄×C₆₇	#41	1	1	yes	characteristic
148	C₂²×C₄×C₆₇	#41	1	1	yes	characteristic
149	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
150	M₁₆×C₆₇	—	2	2	no	—
151	M₁₆×C₆₇	—	2	2	no	—
152	C₂×C₈×C₆₇	—	2	1	no	—
153	C₂×C₈×C₆₇	—	2	1	no	—
154	C₂×C₈×C₆₇	—	2	1	no	—
155	C₂×C₈×C₆₇	—	2	1	no	—
156	C₂×C₈×C₆₇	—	2	2	no	—
157	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
158	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
159	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
160	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
161	C₄⋊C₄×C₆₇	—	2	2	no	—
162	C₄⋊C₄×C₆₇	—	2	2	no	—
163	C₄⋊C₄×C₆₇	—	2	1	no	—
164	C₄⋊C₄×C₆₇	—	2	1	no	—
165	C₄⋊C₄×C₆₇	C₂²×C₄	1	2	yes	—
166	C₄⋊C₄×C₆₇	C₂²×C₄	1	2	yes	—
167	C₂²⋊C₄×C₆₇	—	2	2	no	—
168	C₄²×C₆₇	—	2	2	no	—
169	C₄²×C₆₇	C₂²×C₄	1	1	yes	characteristic
170	C₄²×C₆₇	C₂²×C₄	1	1	yes	characteristic
171	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
172	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
173	C₂³×C₆₇	C₂×#41	1	1	yes	characteristic
174	C₂²×D₆₇	—	2	1	no	—
175	Q₈×C₆₇	—	4	2	no	—
176	Q₈×C₆₇	—	4	2	no	—
177	C₂×C₄×C₆₇	—	2	2	no	—
178	C₂×C₄×C₆₇	—	2	2	no	—
179	C₂×C₄×C₆₇	—	2	2	no	—
180	C₂×C₄×C₆₇	—	4	1	no	—
181	C₂×C₄×C₆₇	C₄×D₄	1	1	yes	characteristic
182	C₂×C₄×C₆₇	C₄×D₄	1	1	yes	characteristic
183	C₂×C₄×C₆₇	C₄×D₄	1	1	yes	characteristic
184	C₂×C₄×C₆₇	C₄×D₄	1	1	yes	characteristic
185	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
186	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
187	C₂×C₄×C₆₇	H(5,𝔽₂)	1	1	yes	characteristic
188	C₂×C₄×C₆₇	2₋¹⁺⁴	1	1	yes	characteristic
189	C₂×C₄×C₆₇	2₋¹⁺⁴	1	1	yes	characteristic
190	C₂×C₄×C₆₇	2₋¹⁺⁴	1	1	yes	characteristic
191	C₂²⋊D₆₇	—	4	1	no	—
192	C₂²⋊D₆₇	—	4	1	no	—
193	C₂×Dic₆₇	—	2	4	no	—
194	C₂×Dic₆₇	—	2	1	no	—
195	C₂×Dic₆₇	—	2	1	no	—
196	C₂×Dic₆₇	—	2	1	no	—
197	C₂×Dic₆₇	—	4	2	no	—
198	C₂×Dic₆₇	—	4	2	no	—
199	D₂₆₈	—	2	1	no	—
200	D₂₆₈	—	2	1	no	—
201	C₄×D₆₇	—	4	2	no	—
202	C₄×D₆₇	—	4	1	no	—
203	C₄×D₆₇	—	4	1	no	—
204	C₄×D₆₇	—	4	2	no	—
205	Dic₁₃₄	—	2	1	no	—
206	Dic₁₃₄	—	2	1	no	—
207	C₈×C₆₇	—	2	1	no	—
208	C₈×C₆₇	—	2	1	no	—
209	C₈×C₆₇	—	2	1	no	—
210	C₈×C₆₇	—	2	1	no	—
211	C₈×C₆₇	—	4	2	no	—
212	C₆₇⋊C₈	—	2	2	no	—
213	C₆₇⋊C₈	—	2	2	no	—
214	C₆₇⋊C₈	—	4	1	no	—
215	C₆₇⋊C₈	—	4	1	no	—
216	C₂²×C₆₇	—	2	1	no	—
217	C₂²×C₆₇	—	2	1	no	—
218	C₂²×C₆₇	C₂×C₄×D₄	1	1	yes	characteristic
219	C₂²×C₆₇	#541	1	1	yes	characteristic
220	C₂²×C₆₇	#542	1	1	yes	characteristic
221	C₂×D₆₇	—	4	1	no	—
222	C₂×D₆₇	—	4	1	no	—
223	C₂×D₆₇	—	4	1	no	—
224	C₄×C₆₇	—	2	1	no	—
225	C₄×C₆₇	—	2	1	no	—
226	C₄×C₆₇	—	2	1	no	—
227	C₄×C₆₇	—	4	2	no	—
228	C₄×C₆₇	—	4	2	no	—
229	C₄×C₆₇	#443	1	2	yes	—
230	C₄×C₆₇	#519	1	1	yes	characteristic
231	C₄×C₆₇	#520	1	1	yes	characteristic
232	C₄×C₆₇	#536	1	1	yes	characteristic
233	C₄×C₆₇	#537	1	1	yes	characteristic
234	Dic₆₇	—	4	4	no	—
235	Dic₆₇	—	4	1	no	—
236	Dic₆₇	—	4	1	no	—
237	Dic₆₇	—	4	1	no	—
238	#22130	—	67	1	no	maximal, Sylow
239	C₂×C₆₇	—	2	1	no	—
240	C₂×C₆₇	—	2	1	no	—
241	C₂×C₆₇	#2323	1	1	yes	characteristic
242	C₂×C₆₇	C₂×#443	1	1	yes	characteristic
243	C₂×C₆₇	#2955	1	1	yes	characteristic
244	D₆₇	—	8	1	no	—
245	#3076	—	67	1	no	—
246	#2960	D₆₇	1	1	yes	characteristic, complemented
247	#2922	—	67	1	no	—
248	#1986	—	67	1	no	—
249	#1950	—	67	1	no	—
250	#1945	—	67	2	no	—
251	#1937	—	67	2	no	—
252	#1929	—	67	2	no	—
253	#1922	—	67	2	no	—
254	#1806	—	67	1	no	—
255	C₈×C₄⋊C₄	—	67	1	no	—
256	C₆₇	#22130	1	1	yes	characteristic, complemented, Sylow
257	#533	C₂×D₆₇	1	1	yes	characteristic
258	C₄×#41	—	67	1	no	—
259	#479	—	67	2	no	—
260	#478	—	67	2	no	—
261	#477	—	67	1	no	—
262	#476	—	67	1	no	—
263	#475	—	67	1	no	—
264	#474	—	67	1	no	—
265	#446	—	2	1	no	—
266	#446	—	2	1	no	—
267	C₈×Q₈	—	2	1	no	—
268	C₈×Q₈	—	2	1	no	—
269	#433	—	134	1	no	—
270	#433	—	134	1	no	—
271	#431	—	134	1	no	—
272	#431	—	134	1	no	—
273	C₂×#108	—	67	2	no	—
274	#427	—	67	2	no	—
275	#424	—	67	1	no	—
276	#424	—	67	1	no	—
277	#424	C₂×D₆₇	1	2	yes	—
278	#423	C₂×D₆₇	1	1	yes	characteristic
279	C₂×C₄⋊C₈	—	67	1	no	—
280	C₂×#104	—	67	2	no	—
281	#416	—	67	2	no	—
282	#408	—	67	1	no	—
283	#408	—	67	1	no	—
284	#405	C₂×D₆₇	1	2	yes	—
285	C₄×M₁₆	—	67	1	no	—
286	C₂×C₄×C₈	—	67	2	no	—
287	C₂×C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
288	C₄×C₄⋊C₄	—	67	1	no	—
289	#341	—	67	2	no	—
290	#339	—	67	2	no	—
291	#339	—	67	2	no	—
292	#339	—	67	2	no	—
293	#336	—	67	2	no	—
294	C₂×#41	—	67	1	no	—
295	C₂×M₁₆	—	67	1	no	—
296	C₂×M₁₆	—	67	1	no	—
297	C₂×M₁₆	C₂²×D₆₇	1	2	yes	—
298	C₂²×C₈	—	67	2	no	—
299	C₂²×C₈	C₂²×D₆₇	1	1	yes	characteristic
300	C₂²×C₈	C₂²×D₆₇	1	1	yes	characteristic
301	C₄⋊Q₈	C₄×D₆₇	1	1	yes	characteristic
302	C₄⋊Q₈	C₄×D₆₇	1	1	yes	characteristic
303	#125	C₄×D₆₇	1	2	yes	—
304	C₂²⋊Q₈	—	2	1	no	—
305	C₂²⋊Q₈	—	2	1	no	—
306	C₄×Q₈	—	2	1	no	—
307	C₄×Q₈	—	2	1	no	—
308	C₄×Q₈	—	67	1	no	—
309	C₄×Q₈	—	67	1	no	—
310	C₄×D₄	—	67	1	no	—
311	C₄×D₄	—	67	1	no	—
312	C₄×D₄	—	134	1	no	—
313	C₄×D₄	—	134	1	no	—
314	#117	—	67	1	no	—
315	#117	—	134	1	no	—
316	#117	C₂²×D₆₇	1	2	yes	—
317	C₂×C₄⋊C₄	—	134	1	no	—
318	C₂×C₄²	—	67	2	no	—
319	C₂×C₄²	—	134	1	no	—
320	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
321	#108	—	268	2	no	—
322	#107	—	67	2	no	—
323	#107	—	67	2	no	—
324	#106	—	67	2	no	—
325	#106	—	67	2	no	—
326	C₄⋊C₈	—	2	2	no	—
327	C₄⋊C₈	—	2	2	no	—
328	C₄⋊C₈	—	2	1	no	—
329	C₄⋊C₈	—	2	1	no	—
330	C₄⋊C₈	—	67	1	no	—
331	C₄⋊C₈	—	67	1	no	—
332	C₄⋊C₈	—	67	2	no	—
333	C₄⋊C₈	—	134	1	no	—
334	C₄⋊C₈	—	134	1	no	—
335	C₄⋊C₈	—	134	1	no	—
336	C₄⋊C₈	—	134	1	no	—
337	#104	—	268	2	no	—
338	#103	—	67	2	no	—
339	#103	—	67	2	no	—
340	#102	—	67	2	no	—
341	#102	—	67	2	no	—
342	C₂²⋊C₈	—	134	1	no	—
343	C₂²⋊C₈	—	134	1	no	—
344	C₂²⋊C₈	—	134	1	no	—
345	C₂²⋊C₈	—	134	1	no	—
346	#97	—	2	2	no	—
347	#97	—	134	1	no	—
348	C₄×C₈	—	2	2	no	—
349	C₄×C₈	—	268	2	no	—
350	C₄×C₈	—	134	1	no	—
351	C₄×C₈	C₂²⋊D₆₇	1	1	yes	characteristic
352	C₄×C₈	C₂²⋊D₆₇	1	1	yes	characteristic
353	C₄×C₈	C₂²⋊D₆₇	1	1	yes	characteristic
354	C₄×C₈	C₂²⋊D₆₇	1	1	yes	characteristic
355	#95	—	134	1	no	—
356	#41	—	134	1	no	—
357	#41	—	134	1	no	—
358	#41	—	134	1	no	—
359	#41	—	134	1	no	—
360	C₂×Q₈	—	2	1	no	—
361	C₂×Q₈	—	2	1	no	—
362	C₂×Q₈	—	67	1	no	—
363	C₂×D₄	—	67	1	no	—
364	C₂×D₄	—	134	1	no	—
365	C₂²×C₄	—	134	2	no	—
366	C₂²×C₄	—	134	1	no	—
367	C₂²×C₄	—	134	1	no	—
368	C₂²×C₄	—	134	1	no	—
369	C₂²×C₄	#11761628	1	1	yes	characteristic
370	C₂²×C₄	#11761628	1	1	yes	characteristic
371	C₂²×C₄	C₂³×D₆₇	1	1	yes	characteristic
372	M₁₆	—	2	2	no	—
373	M₁₆	—	2	2	no	—
374	M₁₆	—	134	1	no	—
375	M₁₆	—	134	1	no	—
376	M₁₆	—	134	1	no	—
377	M₁₆	—	134	1	no	—
378	C₂×C₈	—	2	1	no	—
379	C₂×C₈	—	2	1	no	—
380	C₂×C₈	—	2	1	no	—
381	C₂×C₈	—	2	1	no	—
382	C₂×C₈	—	2	2	no	—
383	C₂×C₈	—	67	2	no	—
384	C₂×C₈	—	67	2	no	—
385	C₂×C₈	—	134	1	no	—
386	C₂×C₈	—	134	1	no	—
387	C₂×C₈	—	134	2	no	—
388	C₂×C₈	—	134	2	no	—
389	C₂×C₈	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
390	C₂×C₈	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
391	C₂×C₈	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
392	C₂×C₈	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
393	C₄⋊C₄	—	2	2	no	—
394	C₄⋊C₄	—	2	2	no	—
395	C₄⋊C₄	—	2	1	no	—
396	C₄⋊C₄	—	2	1	no	—
397	C₄⋊C₄	—	67	1	no	—
398	C₄⋊C₄	—	67	1	no	—
399	C₄⋊C₄	—	134	4	no	—
400	C₄⋊C₄	—	134	1	no	—
401	C₄⋊C₄	—	134	1	no	—
402	C₄⋊C₄	C₂×C₄×D₆₇	1	2	yes	—
403	C₄⋊C₄	C₂×C₄×D₆₇	1	2	yes	—
404	C₂²⋊C₄	—	2	2	no	—
405	C₂²⋊C₄	—	134	1	no	—
406	C₂²⋊C₄	—	134	1	no	—
407	C₂²⋊C₄	—	134	1	no	—
408	C₄²	—	2	2	no	—
409	C₄²	—	134	2	no	—
410	C₄²	—	134	2	no	—
411	C₄²	—	134	1	no	—
412	C₄²	—	134	1	no	—
413	C₄²	—	134	1	no	—
414	C₄²	C₂×C₄×D₆₇	1	1	yes	characteristic
415	C₄²	C₂×C₄×D₆₇	1	1	yes	characteristic
416	C₄²	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
417	C₄²	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
418	C₂³	—	134	1	no	—
419	C₂³	C₂×#11761628	1	1	yes	characteristic
420	Q₈	—	4	2	no	—
421	Q₈	—	4	2	no	—
422	Q₈	—	134	1	no	—
423	Q₈	—	134	1	no	—
424	D₄	—	268	1	no	—
425	D₄	—	268	1	no	—
426	D₄	—	134	1	no	—
427	D₄	—	134	1	no	—
428	C₂×C₄	—	2	2	no	—
429	C₂×C₄	—	2	2	no	—
430	C₂×C₄	—	2	2	no	—
431	C₂×C₄	—	4	1	no	—
432	C₂×C₄	—	268	2	no	—
433	C₂×C₄	—	268	2	no	—
434	C₂×C₄	—	268	2	no	—
435	C₂×C₄	—	268	2	no	—
436	C₂×C₄	—	268	1	no	—
437	C₂×C₄	—	268	1	no	—
438	C₂×C₄	—	134	4	no	—
439	C₂×C₄	—	134	1	no	—
440	C₂×C₄	—	134	1	no	—
441	C₂×C₄	—	134	1	no	—
442	C₂×C₄	C₄×C₂²⋊D₆₇	1	1	yes	characteristic
443	C₂×C₄	C₄×C₂²⋊D₆₇	1	1	yes	characteristic
444	C₂×C₄	C₄×C₂²⋊D₆₇	1	1	yes	characteristic
445	C₂×C₄	C₄×C₂²⋊D₆₇	1	1	yes	characteristic
446	C₂×C₄	C₂²×C₄×D₆₇	1	1	yes	characteristic
447	C₂×C₄	C₂²×C₂²⋊D₆₇	1	1	yes	characteristic
448	C₂×C₄	#14524089	1	1	yes	characteristic
449	C₂×C₄	#14524090	1	1	yes	characteristic
450	C₂×C₄	#14524091	1	1	yes	characteristic
451	C₂×C₄	#14524091	1	1	yes	characteristic
452	C₈	—	2	1	no	—
453	C₈	—	2	1	no	—
454	C₈	—	2	1	no	—
455	C₈	—	2	1	no	—
456	C₈	—	4	2	no	—
457	C₈	—	268	1	no	—
458	C₈	—	268	1	no	—
459	C₈	—	134	2	no	—
460	C₈	—	134	2	no	—
461	C₂²	—	2	1	no	—
462	C₂²	—	2	1	no	—
463	C₂²	—	268	1	no	—
464	C₂²	—	268	1	no	—
465	C₂²	—	268	1	no	—
466	C₂²	C₂×C₄×C₂²⋊D₆₇	1	1	yes	characteristic
467	C₂²	#16864004	1	1	yes	characteristic
468	C₂²	#16864018	1	1	yes	characteristic
469	C₄	—	2	1	no	—
470	C₄	—	2	1	no	—
471	C₄	—	2	1	no	—
472	C₄	—	4	2	no	—
473	C₄	—	4	2	no	—
474	C₄	—	268	4	no	—
475	C₄	—	268	1	no	—
476	C₄	—	268	1	no	—
477	C₄	—	268	1	no	—
478	C₄	#16863511	1	2	yes	—
479	C₄	#16863872	1	1	yes	characteristic
480	C₄	#16863878	1	1	yes	characteristic
481	C₄	#16863970	1	1	yes	characteristic
482	C₄	#16863974	1	1	yes	characteristic
483	C₂	—	2	1	no	—
484	C₂	—	2	1	no	—
485	C₂	—	536	1	no	—
486	C₂	#22562885	1	1	yes	characteristic
487	C₂	C₂×#16863511	1	1	yes	characteristic
488	C₂	#22569110	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	C₆₇×#2960	C₂
socle	C₂²×C₆₇	C₂×C₄×D₄
prime radical	C₂×#11761628	C₂³
cosocle	C₂²×C₄×C₆₇	C₂⁴
hypercenter	#2960	D₆₇
hypocenter	C₆₇	#22130
LCS	G ▷ C₂×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄×C₈ ◁ #2960

Complements

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

N	H ≅ G/N	type
3. #22561154	C₂	semidirect
4. #22560982	C₂	semidirect
5. #22560954	C₂	semidirect
8. #22560808	C₂	semidirect
10. C₈×C₄⋊Dic₆₇	C₂	semidirect
11. C₆₇×#2960	C₂	semidirect
13. #16863700	C₄	semidirect
14. #16863693	C₄	semidirect
15. #16863688	C₄	semidirect
16. #16863683	C₄	semidirect
17. #16863678	C₄	semidirect
18. #16863673	C₄	semidirect
246. #2960	D₆₇	semidirect
256. C₆₇	#22130	semidirect

Maximal subgroups

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

subgroup	index
#22569991	2
#22568851	2
#22561154	2
#22560982	2
#22560954	2
#22560912	2
#22560857	2
#22560808	2
#22560226	2
C₈×C₄⋊Dic₆₇	2
C₆₇×#2960	2
#22130	67