Subgroups — #30825079

Order 17152.

634 conjugacy classes of subgroups.

The subgroup lattice has 412 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22569175	C₂	1	1	yes	maximal, characteristic
2	#22568874	C₂	1	1	yes	maximal, characteristic
3	#22568774	C₂	1	1	yes	maximal, characteristic
4	#22558996	C₂	1	2	yes	maximal, complemented
5	#22558991	C₂	1	2	yes	maximal, complemented
6	#22558983	C₂	1	1	yes	maximal, characteristic, complemented
7	#22558979	C₂	1	1	yes	maximal, characteristic, complemented
8	#22558973	C₂	1	2	yes	maximal, complemented
9	#22558968	C₂	1	1	yes	maximal, characteristic, complemented
10	#22558958	C₂	1	2	yes	maximal, complemented
11	C₂×#334×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#14524049	—	2	1	no	—
13	C₄×#11761628	C₂²	1	1	yes	characteristic
14	#16863526	C₄	1	1	yes	characteristic
15	#16863526	C₄	1	1	yes	characteristic
16	#16863475	C₄	1	1	yes	characteristic
17	#16863475	C₄	1	1	yes	characteristic
18	#16863467	C₄	1	2	yes	—
19	#16863467	C₄	1	2	yes	—
20	#16863456	—	2	2	no	—
21	#16863452	C₂²	1	2	yes	—
22	#16863449	C₂²	1	1	yes	characteristic
23	#16863448	C₂²	1	2	yes	—
24	#16863447	C₂²	1	2	yes	—
25	#16863422	—	2	1	no	—
26	#16863421	—	2	2	no	—
27	#16863420	C₂²	1	1	yes	characteristic
28	#16863420	C₂²	1	1	yes	characteristic
29	#16863418	C₂²	1	2	yes	—
30	C₄⋊C₈×D₆₇	C₂²	1	2	yes	—
31	#16863362	—	2	1	no	—
32	#16863351	—	2	1	no	—
33	C₄×#11761611	C₂²	1	2	yes	—
34	#16863149	C₂²	1	2	yes	—
35	#16863149	C₂²	1	2	yes	—
36	#16863148	C₂²	1	2	yes	—
37	#16863148	C₂²	1	2	yes	—
38	#16863146	C₂²	1	2	yes	—
39	#16863146	C₂²	1	2	yes	—
40	C₂×C₄⋊C₈×C₆₇	C₂²	1	1	yes	characteristic
41	C₂×C₄⋊C₈×C₆₇	C₂²	1	1	yes	characteristic
42	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
43	#334×C₆₇	C₂²	1	1	yes	characteristic
44	#334×C₆₇	C₂²	1	1	yes	characteristic
45	#334×C₆₇	C₂²	1	1	yes	characteristic
46	#334×C₆₇	C₂²	1	1	yes	characteristic
47	C₂×#11761628	D₄	1	1	yes	characteristic
48	#14524049	—	4	4	no	—
49	C₂×#11761612	—	2	1	no	—
50	C₂×#11761612	—	2	1	no	—
51	C₂×#11761611	—	2	2	no	—
52	C₂×#11761611	C₂×C₄	1	2	yes	—
53	C₂×#11761611	C₂×C₄	1	2	yes	—
54	C₂×C₈×D₆₇	—	2	2	no	—
55	C₄×Dic₁₃₄	C₂×C₄	1	1	yes	characteristic
56	C₄×Dic₁₃₄	Q₈	1	1	yes	characteristic
57	C₄×D₂₆₈	C₂×C₄	1	1	yes	characteristic
58	C₄×D₂₆₈	Q₈	1	1	yes	characteristic
59	C₄×C₂²⋊D₆₇	C₂×C₄	1	2	yes	—
60	C₄×C₂²⋊D₆₇	Q₈	1	2	yes	—
61	#14524002	D₄	1	1	yes	characteristic
62	#14524000	D₄	1	2	yes	—
63	C₄²×D₆₇	C₂³	1	2	yes	—
64	C₄⋊(C₆₇⋊C₈)	—	2	1	no	—
65	C₄⋊(C₆₇⋊C₈)	—	2	1	no	—
66	C₄⋊(C₆₇⋊C₈)	C₂×C₄	1	1	yes	characteristic
67	C₄⋊(C₆₇⋊C₈)	C₂×C₄	1	1	yes	characteristic
68	C₄⋊(C₆₇⋊C₈)	C₂³	1	2	yes	—
69	C₄⋊(C₆₇⋊C₈)	C₂³	1	2	yes	—
70	Dic₆₇⋊C₈	—	2	2	no	—
71	Dic₆₇⋊C₈	—	2	2	no	—
72	Dic₆₇⋊C₈	C₂×C₄	1	2	yes	—
73	Dic₆₇⋊C₈	C₂×C₄	1	2	yes	—
74	C₂²⋊(C₆₇⋊C₈)	C₂×C₄	1	2	yes	—
75	(C₂×D₆₇)⋊C₈	—	2	2	no	—
76	(C₂×D₆₇)⋊C₈	—	2	2	no	—
77	(C₂×D₆₇)⋊C₈	C₂×C₄	1	2	yes	—
78	(C₂×D₆₇)⋊C₈	C₂×C₄	1	2	yes	—
79	#14523953	—	2	2	no	—
80	#14523953	C₂×C₄	1	2	yes	—
81	#14523953	C₂×C₄	1	2	yes	—
82	C₄×C₆₇⋊C₈	C₂³	1	2	yes	—
83	C₈×Dic₆₇	—	2	2	no	—
84	C₂²×C₈×C₆₇	—	2	1	no	—
85	C₂²×C₈×C₆₇	—	2	1	no	—
86	C₂²×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
87	C₂²×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
88	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
89	C₄⋊C₈×C₆₇	—	2	1	no	—
90	C₄⋊C₈×C₆₇	—	2	1	no	—
91	C₄⋊C₈×C₆₇	C₂³	1	1	yes	characteristic
92	C₄⋊C₈×C₆₇	C₂³	1	1	yes	characteristic
93	C₄⋊C₈×C₆₇	C₂³	1	1	yes	characteristic
94	C₄⋊C₈×C₆₇	C₂³	1	1	yes	characteristic
95	C₄×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
96	C₄×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
97	C₄×C₈×C₆₇	C₂³	1	1	yes	characteristic
98	C₄×C₈×C₆₇	C₂³	1	1	yes	characteristic
99	#11761628	—	2	2	no	—
100	#11761628	—	2	2	no	—
101	C₂×Dic₁₃₄	C₄⋊C₄	1	1	yes	characteristic
102	C₂×D₂₆₈	C₄⋊C₄	1	1	yes	characteristic
103	C₂×C₂²⋊D₆₇	C₄⋊C₄	1	2	yes	—
104	C₂×C₄×D₆₇	C₂²×C₄	1	2	yes	—
105	C₂×C₄×D₆₇	C₂×D₄	1	2	yes	—
106	C₂×C₄×D₆₇	C₂×Q₈	1	2	yes	—
107	#11761612	—	4	2	no	—
108	#11761612	—	4	2	no	—
109	#11761611	—	2	4	no	—
110	#11761611	—	4	2	no	—
111	#11761611	—	4	4	no	—
112	C₂×C₆₇⋊C₈	—	2	2	no	—
113	C₂×C₆₇⋊C₈	—	2	2	no	—
114	C₂×C₆₇⋊C₈	C₂²×C₄	1	2	yes	—
115	C₂×C₆₇⋊C₈	C₂²×C₄	1	2	yes	—
116	C₈×D₆₇	—	4	4	no	—
117	C₄⋊Dic₆₇	C₄⋊C₄	1	1	yes	characteristic
118	C₄⋊Dic₆₇	C₄⋊C₄	1	1	yes	characteristic
119	Dic₆₇⋊C₄	C₄⋊C₄	1	2	yes	—
120	Dic₆₇⋊C₄	C₄⋊C₄	1	2	yes	—
121	C₂²⋊Dic₆₇	C₄⋊C₄	1	2	yes	—
122	(C₂×D₆₇)⋊C₄	C₄⋊C₄	1	2	yes	—
123	(C₂×D₆₇)⋊C₄	C₄⋊C₄	1	2	yes	—
124	C₄×Dic₆₇	C₂²×C₄	1	2	yes	—
125	C₄×Dic₆₇	C₂×D₄	1	2	yes	—
126	C₄×Dic₆₇	C₂×Q₈	1	2	yes	—
127	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
128	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
129	C₂²×C₄×C₆₇	C₂×Q₈	1	1	yes	characteristic
130	C₂×C₈×C₆₇	—	2	2	no	—
131	C₂×C₈×C₆₇	—	2	1	no	—
132	C₂×C₈×C₆₇	—	2	1	no	—
133	C₂×C₈×C₆₇	—	2	1	no	—
134	C₂×C₈×C₆₇	—	2	1	no	—
135	C₂×C₈×C₆₇	—	4	1	no	—
136	C₂×C₈×C₆₇	—	4	2	no	—
137	C₂×C₈×C₆₇	—	4	2	no	—
138	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
139	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
140	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
141	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
142	C₄²×C₆₇	C₂²×C₄	1	1	yes	characteristic
143	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
144	C₄²×C₆₇	C₂×Q₈	1	1	yes	characteristic
145	C₄²×C₆₇	C₂⁴	1	1	yes	characteristic
146	C₂³×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
147	C₂²×D₆₇	C₂×C₄⋊C₄	1	2	yes	—
148	C₂×C₄×C₆₇	—	2	1	no	—
149	C₂×C₄×C₆₇	—	2	1	no	—
150	C₂×C₄×C₆₇	—	4	1	no	—
151	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
152	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
153	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
154	C₂×C₄×C₆₇	AGL(1,ℤ/8)	1	2	yes	—
155	C₂×C₄×C₆₇	#137	1	2	yes	—
156	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
157	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
158	C₂×C₄×C₆₇	C₂²×Q₈	1	1	yes	characteristic
159	C₂²⋊D₆₇	—	2	4	no	—
160	C₂×Dic₆₇	C₂×C₄⋊C₄	1	2	yes	—
161	C₂×Dic₆₇	C₂×C₄⋊C₄	1	2	yes	—
162	C₂×Dic₆₇	C₂×C₄⋊C₄	1	2	yes	—
163	D₂₆₈	—	2	2	no	—
164	C₄×D₆₇	—	2	2	no	—
165	C₄×D₆₇	—	2	2	no	—
166	C₄×D₆₇	—	2	2	no	—
167	C₄×D₆₇	—	2	2	no	—
168	C₄×D₆₇	—	4	2	no	—
169	Dic₁₃₄	—	2	2	no	—
170	C₈×C₆₇	—	2	1	no	—
171	C₈×C₆₇	—	2	1	no	—
172	C₈×C₆₇	—	2	1	no	—
173	C₈×C₆₇	—	4	1	no	—
174	C₈×C₆₇	—	4	1	no	—
175	C₈×C₆₇	—	4	1	no	—
176	C₈×C₆₇	—	4	1	no	—
177	C₈×C₆₇	#131	1	2	yes	complemented
178	C₆₇⋊C₈	—	2	2	no	—
179	C₆₇⋊C₈	—	2	2	no	—
180	C₆₇⋊C₈	—	4	2	no	—
181	C₆₇⋊C₈	—	4	2	no	—
182	C₂²×C₆₇	—	2	1	no	—
183	C₂²×C₆₇	—	2	1	no	—
184	C₂²×C₆₇	#428	1	2	yes	—
185	C₂²×C₆₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
186	C₂×D₆₇	—	2	2	no	—
187	C₂×D₆₇	—	2	2	no	—
188	C₂×D₆₇	—	2	2	no	—
189	C₄×C₆₇	—	2	1	no	—
190	C₄×C₆₇	—	2	1	no	—
191	C₄×C₆₇	—	2	1	no	—
192	C₄×C₆₇	#428	1	2	yes	—
193	C₄×C₆₇	C₂×#131	1	1	yes	characteristic
194	C₄×C₆₇	#568	1	1	yes	characteristic
195	C₄×C₆₇	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
196	C₄×C₆₇	C₂×#137	1	1	yes	characteristic
197	Dic₆₇	—	2	2	no	—
198	Dic₆₇	—	2	2	no	—
199	Dic₆₇	—	2	2	no	—
200	#17830	—	67	1	no	maximal, Sylow
201	C₂×C₆₇	—	2	2	no	—
202	C₂×C₆₇	#2906	1	1	yes	characteristic
203	C₂×C₆₇	C₂×#428	1	1	yes	characteristic
204	C₂×C₆₇	#2916	1	1	yes	characteristic
205	D₆₇	—	4	2	no	—
206	#2965	—	67	1	no	—
207	#2925	—	67	1	no	—
208	#2906	—	67	1	no	—
209	#1574	—	67	2	no	—
210	#1573	—	67	2	no	—
211	#1571	—	67	1	no	—
212	#1570	—	67	1	no	—
213	#1569	—	67	2	no	—
214	#1567	—	67	1	no	—
215	#1565	—	67	2	no	—
216	C₂×#334	D₆₇	1	1	yes	characteristic, complemented
217	C₆₇	#17830	1	1	yes	characteristic, complemented, Sylow
218	C₂×#131	—	134	1	no	—
219	C₄×#41	—	67	1	no	—
220	#446	—	67	1	no	—
221	#446	—	67	1	no	—
222	#436	—	67	1	no	—
223	#436	—	67	1	no	—
224	#435	—	67	2	no	—
225	#435	—	67	2	no	—
226	#433	—	134	2	no	—
227	#432	—	67	2	no	—
228	#431	—	67	1	no	—
229	#431	—	67	2	no	—
230	#431	—	67	2	no	—
231	#424	—	134	2	no	—
232	#424	—	134	1	no	—
233	#423	—	67	1	no	—
234	#423	—	67	1	no	—
235	#423	—	67	2	no	—
236	C₂×C₄⋊C₈	—	67	2	no	—
237	C₂×C₄⋊C₈	C₂×D₆₇	1	1	yes	characteristic
238	C₂×C₄⋊C₈	C₂×D₆₇	1	1	yes	characteristic
239	#408	—	134	1	no	—
240	#405	—	134	1	no	—
241	C₄×M₁₆	—	67	2	no	—
242	C₂×C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
243	#335	—	67	2	no	—
244	#335	—	67	2	no	—
245	#335	—	67	2	no	—
246	#335	—	67	2	no	—
247	#334	—	67	2	no	—
248	#334	—	67	2	no	—
249	#334	C₂×D₆₇	1	1	yes	characteristic
250	#334	C₂×D₆₇	1	1	yes	characteristic
251	#334	C₂×D₆₇	1	1	yes	characteristic
252	#334	C₂×D₆₇	1	1	yes	characteristic
253	C₂×#41	—	67	1	no	—
254	#131	—	268	4	no	—
255	C₂×M₁₆	—	67	2	no	—
256	C₂×M₁₆	—	67	2	no	—
257	C₂×M₁₆	—	134	2	no	—
258	C₂×M₁₆	—	134	1	no	—
259	C₂×M₁₆	—	134	1	no	—
260	C₂²×C₈	—	2	1	no	—
261	C₂²×C₈	—	2	1	no	—
262	C₂²×C₈	—	134	2	no	—
263	C₂²×C₈	C₄×D₆₇	1	1	yes	characteristic
264	C₂²×C₈	C₄×D₆₇	1	1	yes	characteristic
265	C₄×Q₈	—	67	1	no	—
266	C₄×Q₈	—	67	1	no	—
267	C₄×D₄	—	67	2	no	—
268	C₄×D₄	—	67	2	no	—
269	C₄×D₄	—	67	1	no	—
270	C₄×D₄	—	67	1	no	—
271	#117	—	67	1	no	—
272	#117	—	67	2	no	—
273	C₂×C₄²	—	67	2	no	—
274	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
275	C₄⋊C₈	—	2	1	no	—
276	C₄⋊C₈	—	2	1	no	—
277	C₄⋊C₈	—	67	2	no	—
278	C₄⋊C₈	—	67	2	no	—
279	C₄⋊C₈	—	67	1	no	—
280	C₄⋊C₈	—	67	1	no	—
281	C₄⋊C₈	—	67	2	no	—
282	C₄⋊C₈	—	67	2	no	—
283	C₄⋊C₈	—	134	2	no	—
284	C₄⋊C₈	—	134	2	no	—
285	C₄⋊C₈	—	134	1	no	—
286	C₄⋊C₈	—	134	1	no	—
287	C₄⋊C₈	C₂²×D₆₇	1	1	yes	characteristic
288	C₄⋊C₈	C₂²×D₆₇	1	1	yes	characteristic
289	C₄⋊C₈	C₂²×D₆₇	1	1	yes	characteristic
290	C₄⋊C₈	C₂²×D₆₇	1	1	yes	characteristic
291	C₂²⋊C₈	—	67	2	no	—
292	C₂²⋊C₈	—	67	2	no	—
293	C₂²⋊C₈	—	67	2	no	—
294	C₂²⋊C₈	—	134	2	no	—
295	C₂²⋊C₈	—	134	2	no	—
296	#97	—	67	2	no	—
297	#97	—	67	2	no	—
298	#97	—	134	2	no	—
299	C₄×C₈	—	67	2	no	—
300	C₄×C₈	—	134	2	no	—
301	C₄×C₈	C₄×D₆₇	1	1	yes	characteristic
302	C₄×C₈	C₄×D₆₇	1	1	yes	characteristic
303	C₄×C₈	C₂²×D₆₇	1	1	yes	characteristic
304	C₄×C₈	C₂²×D₆₇	1	1	yes	characteristic
305	#41	—	134	2	no	—
306	#41	—	134	2	no	—
307	C₂×Q₈	—	67	1	no	—
308	C₂×D₄	—	67	2	no	—
309	C₂×D₄	—	67	1	no	—
310	C₂²×C₄	—	67	2	no	—
311	C₂²×C₄	—	67	2	no	—
312	C₂²×C₄	—	67	2	no	—
313	C₂²×C₄	C₂×C₄×D₆₇	1	1	yes	characteristic
314	C₂²×C₄	D₄×D₆₇	1	1	yes	characteristic
315	C₂²×C₄	Q₈×D₆₇	1	1	yes	characteristic
316	M₁₆	—	268	2	no	—
317	M₁₆	—	268	4	no	—
318	M₁₆	—	268	2	no	—
319	M₁₆	—	268	2	no	—
320	M₁₆	—	134	4	no	—
321	C₂×C₈	—	2	2	no	—
322	C₂×C₈	—	2	1	no	—
323	C₂×C₈	—	2	1	no	—
324	C₂×C₈	—	2	1	no	—
325	C₂×C₈	—	2	1	no	—
326	C₂×C₈	—	4	1	no	—
327	C₂×C₈	—	4	2	no	—
328	C₂×C₈	—	4	2	no	—
329	C₂×C₈	—	268	4	no	—
330	C₂×C₈	—	67	2	no	—
331	C₂×C₈	—	67	2	no	—
332	C₂×C₈	—	134	2	no	—
333	C₂×C₈	—	134	2	no	—
334	C₂×C₈	C₂×C₄×D₆₇	1	1	yes	characteristic
335	C₂×C₈	C₂×C₄×D₆₇	1	1	yes	characteristic
336	C₂×C₈	C₂×C₄×D₆₇	1	1	yes	characteristic
337	C₂×C₈	C₂×C₄×D₆₇	1	1	yes	characteristic
338	C₄⋊C₄	—	67	2	no	—
339	C₄⋊C₄	—	67	2	no	—
340	C₄⋊C₄	—	67	1	no	—
341	C₄⋊C₄	—	67	1	no	—
342	C₂²⋊C₄	—	67	2	no	—
343	C₂²⋊C₄	—	67	2	no	—
344	C₂²⋊C₄	—	67	2	no	—
345	C₄²	—	67	2	no	—
346	C₄²	—	67	2	no	—
347	C₄²	—	67	2	no	—
348	C₄²	C₂×C₄×D₆₇	1	1	yes	characteristic
349	C₄²	D₄×D₆₇	1	1	yes	characteristic
350	C₄²	Q₈×D₆₇	1	1	yes	characteristic
351	C₄²	C₂³×D₆₇	1	1	yes	characteristic
352	C₂³	—	67	2	no	—
353	C₂³	C₄⋊C₄×D₆₇	1	1	yes	characteristic
354	Q₈	—	134	2	no	—
355	D₄	—	134	4	no	—
356	D₄	—	134	2	no	—
357	C₂×C₄	—	2	1	no	—
358	C₂×C₄	—	2	1	no	—
359	C₂×C₄	—	4	1	no	—
360	C₂×C₄	—	268	2	no	—
361	C₂×C₄	—	67	2	no	—
362	C₂×C₄	—	67	2	no	—
363	C₂×C₄	—	67	2	no	—
364	C₂×C₄	—	134	2	no	—
365	C₂×C₄	—	134	2	no	—
366	C₂×C₄	—	134	2	no	—
367	C₂×C₄	—	134	2	no	—
368	C₂×C₄	C₄⋊C₄×D₆₇	1	1	yes	characteristic
369	C₂×C₄	C₄⋊C₄×D₆₇	1	1	yes	characteristic
370	C₂×C₄	C₄⋊C₄×D₆₇	1	1	yes	characteristic
371	C₂×C₄	#14524070	1	2	yes	—
372	C₂×C₄	#14524072	1	2	yes	—
373	C₂×C₄	C₂²×C₄×D₆₇	1	1	yes	characteristic
374	C₂×C₄	C₂×D₄×D₆₇	1	1	yes	characteristic
375	C₂×C₄	C₂×Q₈×D₆₇	1	1	yes	characteristic
376	C₈	—	2	1	no	—
377	C₈	—	2	1	no	—
378	C₈	—	2	1	no	—
379	C₈	—	4	1	no	—
380	C₈	—	4	1	no	—
381	C₈	—	4	1	no	—
382	C₈	—	4	1	no	—
383	C₈	—	268	2	no	—
384	C₈	—	268	2	no	—
385	C₈	—	134	2	no	—
386	C₈	—	134	2	no	—
387	C₈	#14524049	1	2	yes	complemented
388	C₂²	—	2	1	no	—
389	C₂²	—	2	1	no	—
390	C₂²	—	134	2	no	—
391	C₂²	—	134	2	no	—
392	C₂²	—	134	2	no	—
393	C₂²	#16863435	1	2	yes	—
394	C₂²	C₂×C₄⋊C₄×D₆₇	1	1	yes	characteristic
395	C₄	—	2	1	no	—
396	C₄	—	2	1	no	—
397	C₄	—	2	1	no	—
398	C₄	—	134	2	no	—
399	C₄	—	134	2	no	—
400	C₄	—	134	2	no	—
401	C₄	#16863435	1	2	yes	—
402	C₄	C₂×#14524049	1	1	yes	characteristic
403	C₄	#16864162	1	1	yes	characteristic
404	C₄	C₂×#14524070	1	1	yes	characteristic
405	C₄	C₂×#14524072	1	1	yes	characteristic
406	C₂	—	2	2	no	—
407	C₂	—	268	2	no	—
408	C₂	#22568774	1	1	yes	characteristic
409	C₂	C₂×#16863435	1	1	yes	characteristic
410	C₂	#22568817	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×D₄×D₆₇
commutator	C₂×C₄×C₆₇	C₂³×C₄
Frattini	C₄²	C₂³×D₆₇
Fitting	C₂×#334×C₆₇	C₂
socle	C₂²×C₆₇	C₂²×C₄⋊C₄
prime radical	C₂×#11761628	D₄
cosocle	C₄²×C₆₇	C₂⁴
hypercenter	C₂×#334	D₆₇
hypocenter	C₆₇	#17830
LCS	G ▷ C₂×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×#334

Complements

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

N	H ≅ G/N	type
4. #22558996	C₂	semidirect
5. #22558991	C₂	semidirect
6. #22558983	C₂	semidirect
7. #22558979	C₂	semidirect
8. #22558973	C₂	semidirect
9. #22558968	C₂	semidirect
10. #22558958	C₂	semidirect
11. C₂×#334×C₆₇	C₂	semidirect
177. C₈×C₆₇	#131	semidirect
216. C₂×#334	D₆₇	semidirect
217. C₆₇	#17830	semidirect
387. C₈	#14524049	semidirect

Maximal subgroups

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

subgroup	index
#22569175	2
#22568874	2
#22568774	2
#22558996	2
#22558991	2
#22558983	2
#22558979	2
#22558973	2
#22558968	2
#22558958	2
C₂×#334×C₆₇	2
#17830	67