Subgroups — #30875286

Order 17152.

752 conjugacy classes of subgroups.

The subgroup lattice has 562 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×#14524063	C₂	1	1	yes	maximal, characteristic
2	#22568875	C₂	1	1	yes	maximal, characteristic, complemented
3	#22568778	C₂	1	1	yes	maximal, characteristic, complemented
4	#22560910	C₂	1	2	yes	maximal, complemented
5	#22560806	C₂	1	2	yes	maximal, complemented
6	#22560533	C₂	1	2	yes	maximal
7	#22560346	C₂	1	1	yes	maximal, characteristic, complemented
8	#22560230	C₂	1	1	yes	maximal, characteristic, complemented
9	C₄⋊C₄×C₆₇⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
10	#22560043	C₂	1	2	yes	maximal, complemented
11	C₆₇×#1806	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#14524063	C₂²	1	1	yes	characteristic
13	C₂×#14524051	C₂²	1	2	yes	—
14	C₄×#11761628	C₂²	1	1	yes	characteristic
15	C₄×#11761618	C₄	1	1	yes	characteristic, complemented
16	C₄×#11761618	C₄	1	1	yes	characteristic, complemented
17	C₄×#11761615	C₄	1	1	yes	characteristic, complemented
18	C₄×#11761615	C₄	1	1	yes	characteristic, complemented
19	C₄×#11761616	C₄	1	1	yes	characteristic, complemented
20	C₄×#11761616	C₄	1	1	yes	characteristic, complemented
21	C₄×#11761613	C₄	1	1	yes	characteristic, complemented
22	C₄×#11761613	C₄	1	1	yes	characteristic, complemented
23	#16863457	—	2	2	no	—
24	#16863453	—	2	1	no	—
25	#16863449	—	2	1	no	—
26	#16863432	C₂²	1	1	yes	characteristic
27	#16863424	C₂²	1	2	yes	—
28	#16863421	—	2	2	no	—
29	#16863420	C₂²	1	1	yes	characteristic
30	#16863418	—	2	1	no	—
31	C₂×C₄⋊(C₆₇⋊C₈)	C₂²	1	1	yes	characteristic
32	C₄⋊C₈×D₆₇	—	2	1	no	—
33	C₂×#14523971	C₂²	1	2	yes	—
34	#16863392	C₂²	1	1	yes	characteristic
35	#16863390	C₂²	1	1	yes	characteristic
36	#16863363	C₂²	1	2	yes	—
37	#16863352	C₂²	1	2	yes	—
38	C₂×C₄×C₆₇⋊C₈	C₂²	1	2	yes	—
39	C₂×C₄×C₆₇⋊C₈	C₂²	1	1	yes	characteristic
40	#16863160	C₂²	1	2	yes	—
41	#16863160	C₂²	1	2	yes	—
42	#16863156	C₂²	1	2	yes	—
43	#16863151	C₂²	1	2	yes	—
44	C₄×#41×C₆₇	C₂²	1	1	yes	characteristic
45	#423×C₆₇	C₂²	1	1	yes	characteristic
46	C₂×#104×C₆₇	C₂²	1	2	yes	—
47	C₄×C₄⋊C₄×C₆₇	C₂²	1	1	yes	characteristic
48	#339×C₆₇	C₂²	1	2	yes	—
49	C₂×#11761628	C₂³	1	1	yes	characteristic
50	#14524063	—	2	2	no	—
51	#14524063	—	2	2	no	—
52	C₂×#11761618	C₂×C₄	1	1	yes	characteristic
53	C₂×#11761616	C₂×C₄	1	1	yes	characteristic
54	C₂×#11761615	C₂×C₄	1	1	yes	characteristic
55	C₂×#11761613	C₂×C₄	1	1	yes	characteristic
56	#14524051	—	2	4	no	—
57	#14524051	—	4	2	no	—
58	C₂×#11761612	C₂³	1	2	yes	—
59	C₂×#11761611	—	2	2	no	—
60	C₂²×C₆₇⋊C₈	C₂³	1	1	yes	characteristic
61	C₂²×C₆₇⋊C₈	C₂³	1	1	yes	characteristic
62	C₂²×C₆₇⋊C₈	C₂³	1	2	yes	—
63	C₂×C₈×D₆₇	—	2	2	no	—
64	C₄×Dic₁₃₄	C₂×C₄	1	1	yes	characteristic
65	C₄×Dic₁₃₄	C₂×C₄	1	1	yes	characteristic
66	C₄×D₂₆₈	C₂×C₄	1	1	yes	characteristic
67	C₄×D₂₆₈	C₂×C₄	1	1	yes	characteristic
68	C₄×C₂²⋊D₆₇	—	2	1	no	—
69	C₄×C₂²⋊D₆₇	—	2	1	no	—
70	#14524002	C₂³	1	1	yes	characteristic
71	#14524000	—	2	1	no	—
72	C₄²×D₆₇	—	2	1	no	—
73	#14523976	C₂×C₄	1	1	yes	characteristic
74	#14523976	C₂×C₄	1	1	yes	characteristic
75	#14523974	C₂×C₄	1	1	yes	characteristic
76	#14523974	C₂×C₄	1	1	yes	characteristic
77	C₄⋊(C₆₇⋊C₈)	—	2	1	no	—
78	C₄⋊(C₆₇⋊C₈)	—	2	1	no	—
79	C₄⋊(C₆₇⋊C₈)	D₄	1	1	yes	characteristic
80	C₄⋊(C₆₇⋊C₈)	D₄	1	1	yes	characteristic
81	C₄⋊(C₆₇⋊C₈)	D₄	1	2	yes	—
82	Dic₆₇⋊C₈	—	2	2	no	—
83	Dic₆₇⋊C₈	—	2	2	no	—
84	#14523971	—	2	4	no	—
85	#14523967	C₂×C₄	1	1	yes	characteristic
86	#14523967	C₂×C₄	1	1	yes	characteristic
87	#14523966	C₂×C₄	1	1	yes	characteristic
88	#14523966	C₂×C₄	1	1	yes	characteristic
89	#14523964	C₂×C₄	1	1	yes	characteristic
90	#14523964	C₂×C₄	1	1	yes	characteristic
91	#14523963	C₂×C₄	1	1	yes	characteristic
92	#14523963	C₂×C₄	1	1	yes	characteristic
93	C₂²⋊(C₆₇⋊C₈)	—	2	2	no	—
94	C₂²⋊(C₆₇⋊C₈)	—	2	2	no	—
95	#14523954	—	2	2	no	—
96	C₄×C₆₇⋊C₈	—	2	2	no	—
97	C₄×C₆₇⋊C₈	—	4	2	no	—
98	C₄×C₆₇⋊C₈	C₂×C₄	1	1	yes	characteristic
99	C₄×C₆₇⋊C₈	C₂×C₄	1	1	yes	characteristic
100	C₄×C₆₇⋊C₈	C₂×C₄	1	1	yes	characteristic
101	C₄×C₆₇⋊C₈	C₂×C₄	1	1	yes	characteristic
102	C₂×#41×C₆₇	C₂³	1	1	yes	characteristic
103	C₂×M₁₆×C₆₇	C₂³	1	2	yes	—
104	C₄×Q₈×C₆₇	C₂×C₄	1	1	yes	characteristic
105	C₄×Q₈×C₆₇	C₂×C₄	1	1	yes	characteristic
106	C₄×D₄×C₆₇	—	2	1	no	—
107	C₄×D₄×C₆₇	—	2	1	no	—
108	C₄×D₄×C₆₇	C₂×C₄	1	1	yes	characteristic
109	C₄×D₄×C₆₇	C₂×C₄	1	1	yes	characteristic
110	#117×C₆₇	—	2	1	no	—
111	#117×C₆₇	C₂³	1	1	yes	characteristic
112	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
113	C₂×C₄²×C₆₇	—	2	1	no	—
114	C₂×C₄²×C₆₇	C₂³	1	2	yes	—
115	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
116	C₄⋊C₈×C₆₇	—	2	1	no	—
117	C₄⋊C₈×C₆₇	—	2	1	no	—
118	#104×C₆₇	—	4	2	no	—
119	#95×C₆₇	—	2	1	no	—
120	#11761628	—	2	1	no	—
121	#11761628	—	2	1	no	—
122	#11761628	#41	1	2	yes	—
123	#11761628	#41	1	2	yes	—
124	C₂×Dic₁₃₄	C₂²×C₄	1	1	yes	characteristic
125	C₂×D₂₆₈	C₂²×C₄	1	1	yes	characteristic
126	C₂×C₂²⋊D₆₇	—	2	1	no	—
127	C₂×C₄×D₆₇	—	2	1	no	—
128	C₂×C₄×D₆₇	—	2	1	no	—
129	C₂×C₄×D₆₇	—	2	1	no	—
130	#11761618	—	2	2	no	—
131	#11761616	—	2	2	no	—
132	#11761615	—	2	2	no	—
133	#11761613	—	2	2	no	—
134	#11761612	—	2	2	no	—
135	#11761612	—	2	2	no	—
136	#11761611	—	4	4	no	—
137	C₂×C₆₇⋊C₈	—	2	2	no	—
138	C₂×C₆₇⋊C₈	—	2	1	no	—
139	C₂×C₆₇⋊C₈	—	2	1	no	—
140	C₂×C₆₇⋊C₈	—	2	1	no	—
141	C₂×C₆₇⋊C₈	—	2	2	no	—
142	C₂×C₆₇⋊C₈	—	2	1	no	—
143	C₂×C₆₇⋊C₈	—	2	2	no	—
144	C₂×C₆₇⋊C₈	C₂²×C₄	1	1	yes	characteristic
145	C₂×C₆₇⋊C₈	C₂²×C₄	1	1	yes	characteristic
146	C₂×C₆₇⋊C₈	C₂²×C₄	1	1	yes	characteristic
147	C₂×C₆₇⋊C₈	C₂²×C₄	1	1	yes	characteristic
148	C₂×C₆₇⋊C₈	C₂×D₄	1	2	yes	—
149	C₂×C₆₇⋊C₈	C₂×D₄	1	2	yes	—
150	C₈×D₆₇	—	4	4	no	—
151	C₄⋊Dic₆₇	C₂²×C₄	1	1	yes	characteristic
152	C₄⋊Dic₆₇	C₂²×C₄	1	1	yes	characteristic
153	Dic₆₇⋊C₄	—	2	1	no	—
154	Dic₆₇⋊C₄	—	2	1	no	—
155	C₂²⋊Dic₆₇	—	2	1	no	—
156	(C₂×D₆₇)⋊C₄	—	2	1	no	—
157	(C₂×D₆₇)⋊C₄	—	2	1	no	—
158	C₄×Dic₆₇	—	2	1	no	—
159	C₄×Dic₆₇	—	2	1	no	—
160	C₄×Dic₆₇	—	2	1	no	—
161	#41×C₆₇	—	2	1	no	—
162	#41×C₆₇	—	2	1	no	—
163	#41×C₆₇	—	2	1	no	—
164	#41×C₆₇	—	2	1	no	—
165	C₂×Q₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
166	C₂×D₄×C₆₇	—	2	1	no	—
167	C₂×D₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
168	C₂²×C₄×C₆₇	—	2	2	no	—
169	C₂²×C₄×C₆₇	—	2	1	no	—
170	C₂²×C₄×C₆₇	—	2	1	no	—
171	C₂²×C₄×C₆₇	—	2	1	no	—
172	C₂²×C₄×C₆₇	#41	1	1	yes	characteristic
173	C₂²×C₄×C₆₇	#41	1	1	yes	characteristic
174	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
175	M₁₆×C₆₇	—	2	2	no	—
176	M₁₆×C₆₇	—	2	2	no	—
177	C₂×C₈×C₆₇	—	2	2	no	—
178	C₄⋊C₄×C₆₇	—	2	4	no	—
179	C₄⋊C₄×C₆₇	—	2	1	no	—
180	C₄⋊C₄×C₆₇	—	2	1	no	—
181	C₄⋊C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
182	C₄⋊C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
183	C₂²⋊C₄×C₆₇	—	2	1	no	—
184	C₂²⋊C₄×C₆₇	—	2	1	no	—
185	C₂²⋊C₄×C₆₇	—	2	1	no	—
186	C₄²×C₆₇	—	2	1	no	—
187	C₄²×C₆₇	—	2	1	no	—
188	C₄²×C₆₇	—	2	1	no	—
189	C₄²×C₆₇	—	2	2	no	—
190	C₄²×C₆₇	—	2	2	no	—
191	C₄²×C₆₇	C₂²×C₄	1	1	yes	characteristic
192	C₄²×C₆₇	C₂²×C₄	1	1	yes	characteristic
193	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
194	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
195	C₂³×C₆₇	—	2	1	no	—
196	C₂³×C₆₇	C₂×#41	1	1	yes	characteristic
197	C₂²×D₆₇	—	2	1	no	—
198	Q₈×C₆₇	—	2	1	no	—
199	Q₈×C₆₇	—	2	1	no	—
200	D₄×C₆₇	—	2	1	no	—
201	D₄×C₆₇	—	2	1	no	—
202	D₄×C₆₇	—	4	1	no	—
203	D₄×C₆₇	—	4	1	no	—
204	C₂×C₄×C₆₇	—	2	2	no	—
205	C₂×C₄×C₆₇	—	2	4	no	—
206	C₂×C₄×C₆₇	—	2	1	no	—
207	C₂×C₄×C₆₇	—	2	1	no	—
208	C₂×C₄×C₆₇	—	2	1	no	—
209	C₂×C₄×C₆₇	—	4	1	no	—
210	C₂×C₄×C₆₇	—	4	2	no	—
211	C₂×C₄×C₆₇	—	4	2	no	—
212	C₂×C₄×C₆₇	—	4	1	no	—
213	C₂×C₄×C₆₇	—	4	1	no	—
214	C₂×C₄×C₆₇	—	4	2	no	—
215	C₂×C₄×C₆₇	—	8	1	no	—
216	C₂×C₄×C₆₇	C₄×D₄	1	1	yes	characteristic
217	C₂×C₄×C₆₇	C₄×D₄	1	1	yes	characteristic
218	C₂×C₄×C₆₇	C₄×D₄	1	1	yes	characteristic
219	C₂×C₄×C₆₇	C₄×D₄	1	1	yes	characteristic
220	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
221	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
222	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
223	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
224	C₂×C₄×C₆₇	H(5,𝔽₂)	1	1	yes	characteristic
225	C₂×C₄×C₆₇	2₋¹⁺⁴	1	1	yes	characteristic
226	C₂²⋊D₆₇	—	2	2	no	—
227	C₂²⋊D₆₇	—	2	2	no	—
228	C₂×Dic₆₇	—	2	1	no	—
229	C₂×Dic₆₇	—	2	1	no	—
230	C₂×Dic₆₇	—	2	1	no	—
231	D₂₆₈	—	2	1	no	—
232	D₂₆₈	#117	1	2	yes	—
233	C₄×D₆₇	—	2	2	no	—
234	C₄×D₆₇	—	2	2	no	—
235	C₄×D₆₇	—	4	2	no	—
236	C₄×D₆₇	—	8	1	no	—
237	Dic₁₃₄	—	2	1	no	—
238	Dic₁₃₄	#117	1	2	yes	—
239	C₈×C₆₇	—	4	2	no	—
240	C₆₇⋊C₈	—	2	1	no	—
241	C₆₇⋊C₈	—	2	1	no	—
242	C₆₇⋊C₈	—	2	1	no	—
243	C₆₇⋊C₈	—	2	1	no	—
244	C₆₇⋊C₈	—	2	2	no	—
245	C₆₇⋊C₈	—	2	2	no	—
246	C₆₇⋊C₈	—	4	2	no	—
247	C₂²×C₆₇	—	2	1	no	—
248	C₂²×C₆₇	—	2	1	no	—
249	C₂²×C₆₇	—	4	1	no	—
250	C₂²×C₆₇	—	4	1	no	—
251	C₂²×C₆₇	—	4	1	no	—
252	C₂²×C₆₇	C₂×C₄×D₄	1	1	yes	characteristic
253	C₂²×C₆₇	#552	1	1	yes	characteristic
254	C₂²×C₆₇	#553	1	1	yes	characteristic
255	C₂×D₆₇	—	2	2	no	—
256	C₂×D₆₇	—	4	1	no	—
257	C₂×D₆₇	—	4	1	no	—
258	C₄×C₆₇	—	2	1	no	—
259	C₄×C₆₇	—	2	1	no	—
260	C₄×C₆₇	—	2	1	no	—
261	C₄×C₆₇	—	4	4	no	—
262	C₄×C₆₇	—	4	1	no	—
263	C₄×C₆₇	—	4	1	no	—
264	C₄×C₆₇	—	4	1	no	—
265	C₄×C₆₇	#443	1	2	yes	—
266	C₄×C₆₇	C₂×#117	1	1	yes	characteristic
267	C₄×C₆₇	#520	1	1	yes	characteristic
268	C₄×C₆₇	#546	1	1	yes	characteristic
269	C₄×C₆₇	#548	1	1	yes	characteristic
270	Dic₆₇	—	2	2	no	—
271	Dic₆₇	—	4	1	no	—
272	Dic₆₇	—	4	1	no	—
273	#22783	—	67	1	no	maximal, Sylow
274	C₂×C₆₇	—	2	1	no	—
275	C₂×C₆₇	—	2	1	no	—
276	C₂×C₆₇	—	8	1	no	—
277	C₂×C₆₇	#2352	1	1	yes	characteristic
278	C₂×C₆₇	C₂×#443	1	1	yes	characteristic
279	C₂×C₆₇	#2957	1	1	yes	characteristic
280	D₆₇	—	4	2	no	—
281	C₄×#135	—	67	1	no	—
282	#2925	—	67	1	no	—
283	#2906	—	67	1	no	—
284	#1937	—	67	2	no	—
285	#1922	—	67	2	no	—
286	#1876	—	67	2	no	—
287	#1837	—	67	1	no	—
288	#1806	—	67	1	no	—
289	#1806	D₆₇	1	1	yes	characteristic, complemented
290	C₈×C₄⋊C₄	—	67	1	no	—
291	#1764	—	67	2	no	—
292	C₆₇	#22783	1	1	yes	characteristic, complemented, Sylow
293	C₂×#135	—	67	1	no	—
294	C₂×#131	—	67	2	no	—
295	C₄×#41	—	67	1	no	—
296	C₄×#41	C₂×D₆₇	1	1	yes	characteristic
297	C₄×Q₁₆	—	67	1	no	—
298	C₄×Q₁₆	—	67	1	no	—
299	C₄×SD₁₆	—	67	1	no	—
300	C₄×SD₁₆	—	67	1	no	—
301	C₄×SD₁₆	—	67	1	no	—
302	C₄×SD₁₆	—	67	1	no	—
303	C₄×D₈	—	67	1	no	—
304	C₄×D₈	—	67	1	no	—
305	#433	—	134	2	no	—
306	#432	—	134	1	no	—
307	#431	—	134	1	no	—
308	#427	—	67	1	no	—
309	#424	—	67	2	no	—
310	#424	—	134	2	no	—
311	#423	—	67	1	no	—
312	#423	—	134	1	no	—
313	#423	C₂×D₆₇	1	1	yes	characteristic
314	C₂×C₄⋊C₈	—	67	1	no	—
315	C₂×C₄⋊C₈	—	134	1	no	—
316	C₂×#104	—	67	2	no	—
317	C₂×#104	C₂×D₆₇	1	2	yes	—
318	#416	—	67	1	no	—
319	#416	—	67	1	no	—
320	#408	—	67	2	no	—
321	#405	—	67	2	no	—
322	C₂×C₄×C₈	—	67	2	no	—
323	C₂×C₄×C₈	—	67	1	no	—
324	C₄×C₄⋊C₄	C₂×D₆₇	1	1	yes	characteristic
325	#341	—	67	2	no	—
326	#341	—	67	2	no	—
327	#339	—	67	2	no	—
328	#339	C₂×D₆₇	1	2	yes	—
329	#336	—	67	2	no	—
330	C₂×#41	—	67	1	no	—
331	C₂×#41	C₂²×D₆₇	1	1	yes	characteristic
332	#135	—	134	2	no	—
333	#135	—	134	2	no	—
334	C₂×Q₁₆	—	67	1	no	—
335	C₂×SD₁₆	—	67	1	no	—
336	C₂×SD₁₆	—	67	1	no	—
337	C₂×D₈	—	67	1	no	—
338	#131	—	268	2	no	—
339	#131	—	134	4	no	—
340	C₂×M₁₆	—	67	2	no	—
341	C₂×M₁₆	—	134	2	no	—
342	C₂×M₁₆	C₂²×D₆₇	1	2	yes	—
343	C₂²×C₈	—	67	2	no	—
344	C₂²×C₈	—	67	1	no	—
345	C₂²×C₈	—	67	1	no	—
346	C₂²×C₈	—	134	2	no	—
347	C₄×Q₈	—	67	1	no	—
348	C₄×Q₈	—	67	1	no	—
349	C₄×Q₈	C₄×D₆₇	1	1	yes	characteristic
350	C₄×Q₈	C₄×D₆₇	1	1	yes	characteristic
351	C₄×D₄	—	2	1	no	—
352	C₄×D₄	—	2	1	no	—
353	C₄×D₄	—	67	1	no	—
354	C₄×D₄	—	67	1	no	—
355	C₄×D₄	—	134	1	no	—
356	C₄×D₄	—	134	1	no	—
357	C₄×D₄	C₄×D₆₇	1	1	yes	characteristic
358	C₄×D₄	C₄×D₆₇	1	1	yes	characteristic
359	#117	—	2	1	no	—
360	#117	—	67	1	no	—
361	#117	—	134	1	no	—
362	#117	C₂²×D₆₇	1	1	yes	characteristic
363	C₂×C₄⋊C₄	—	2	1	no	—
364	C₂×C₄²	—	2	1	no	—
365	C₂×C₄²	—	134	1	no	—
366	C₂×C₄²	C₂²×D₆₇	1	2	yes	—
367	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
368	#107	—	67	1	no	—
369	#107	—	67	1	no	—
370	#106	—	67	1	no	—
371	#106	—	67	1	no	—
372	C₄⋊C₈	—	2	1	no	—
373	C₄⋊C₈	—	2	1	no	—
374	C₄⋊C₈	—	67	1	no	—
375	C₄⋊C₈	—	67	1	no	—
376	C₄⋊C₈	—	67	2	no	—
377	C₄⋊C₈	—	134	2	no	—
378	C₄⋊C₈	—	134	2	no	—
379	C₄⋊C₈	—	134	1	no	—
380	C₄⋊C₈	—	134	1	no	—
381	#104	—	4	2	no	—
382	#104	—	134	4	no	—
383	#103	—	67	1	no	—
384	#103	—	67	1	no	—
385	#103	—	67	1	no	—
386	#103	—	67	1	no	—
387	#102	—	67	1	no	—
388	#102	—	67	1	no	—
389	#102	—	67	1	no	—
390	#102	—	67	1	no	—
391	C₂²⋊C₈	—	134	2	no	—
392	C₂²⋊C₈	—	134	2	no	—
393	#97	—	134	2	no	—
394	C₄×C₈	—	268	2	no	—
395	C₄×C₈	—	67	1	no	—
396	C₄×C₈	—	67	1	no	—
397	C₄×C₈	—	67	1	no	—
398	C₄×C₈	—	67	1	no	—
399	C₄×C₈	—	134	2	no	—
400	#95	—	2	1	no	—
401	#41	—	2	1	no	—
402	#41	—	2	1	no	—
403	#41	—	2	1	no	—
404	#41	—	2	1	no	—
405	#41	—	67	2	no	—
406	#41	—	67	2	no	—
407	#41	—	134	1	no	—
408	#41	—	134	1	no	—
409	C₂×Q₈	—	67	1	no	—
410	C₂×Q₈	C₂×C₄×D₆₇	1	1	yes	characteristic
411	C₂×D₄	—	2	1	no	—
412	C₂×D₄	—	67	1	no	—
413	C₂×D₄	—	134	1	no	—
414	C₂×D₄	C₂×C₄×D₆₇	1	1	yes	characteristic
415	C₂²×C₄	—	2	2	no	—
416	C₂²×C₄	—	2	1	no	—
417	C₂²×C₄	—	2	1	no	—
418	C₂²×C₄	—	2	1	no	—
419	C₂²×C₄	—	134	1	no	—
420	C₂²×C₄	—	134	1	no	—
421	C₂²×C₄	—	134	1	no	—
422	C₂²×C₄	#11761627	1	1	yes	characteristic
423	C₂²×C₄	#11761627	1	1	yes	characteristic
424	C₂²×C₄	C₂³×D₆₇	1	1	yes	characteristic
425	Q₁₆	—	134	2	no	—
426	SD₁₆	—	134	2	no	—
427	SD₁₆	—	134	2	no	—
428	D₈	—	134	2	no	—
429	M₁₆	—	2	2	no	—
430	M₁₆	—	2	2	no	—
431	M₁₆	—	268	4	no	—
432	M₁₆	—	134	2	no	—
433	M₁₆	—	134	2	no	—
434	C₂×C₈	—	2	2	no	—
435	C₂×C₈	—	268	4	no	—
436	C₂×C₈	—	67	1	no	—
437	C₂×C₈	—	67	1	no	—
438	C₂×C₈	—	67	1	no	—
439	C₂×C₈	—	67	1	no	—
440	C₂×C₈	—	67	2	no	—
441	C₂×C₈	—	67	2	no	—
442	C₂×C₈	—	134	2	no	—
443	C₂×C₈	—	134	2	no	—
444	C₂×C₈	—	134	1	no	—
445	C₂×C₈	—	134	1	no	—
446	C₂×C₈	—	134	1	no	—
447	C₂×C₈	—	134	2	no	—
448	C₂×C₈	—	134	1	no	—
449	C₄⋊C₄	—	2	4	no	—
450	C₄⋊C₄	—	2	1	no	—
451	C₄⋊C₄	—	2	1	no	—
452	C₄⋊C₄	—	67	1	no	—
453	C₄⋊C₄	—	67	1	no	—
454	C₄⋊C₄	—	134	1	no	—
455	C₄⋊C₄	—	134	1	no	—
456	C₄⋊C₄	C₂×C₄×D₆₇	1	1	yes	characteristic
457	C₄⋊C₄	C₂×C₄×D₆₇	1	1	yes	characteristic
458	C₂²⋊C₄	—	2	1	no	—
459	C₂²⋊C₄	—	2	1	no	—
460	C₂²⋊C₄	—	2	1	no	—
461	C₂²⋊C₄	—	134	1	no	—
462	C₂²⋊C₄	—	134	1	no	—
463	C₂²⋊C₄	—	134	1	no	—
464	C₄²	—	2	1	no	—
465	C₄²	—	2	1	no	—
466	C₄²	—	2	1	no	—
467	C₄²	—	2	2	no	—
468	C₄²	—	2	2	no	—
469	C₄²	—	134	1	no	—
470	C₄²	—	134	1	no	—
471	C₄²	—	134	1	no	—
472	C₄²	C₂×C₄×D₆₇	1	1	yes	characteristic
473	C₄²	C₂×C₄×D₆₇	1	1	yes	characteristic
474	C₄²	D₄×D₆₇	1	1	yes	characteristic
475	C₄²	D₄×D₆₇	1	1	yes	characteristic
476	C₂³	—	2	1	no	—
477	C₂³	—	134	1	no	—
478	C₂³	C₂×#11761627	1	1	yes	characteristic
479	Q₈	—	2	1	no	—
480	Q₈	—	2	1	no	—
481	Q₈	—	67	2	no	—
482	Q₈	—	134	1	no	—
483	D₄	—	2	1	no	—
484	D₄	—	2	1	no	—
485	D₄	—	4	1	no	—
486	D₄	—	4	1	no	—
487	D₄	—	67	2	no	—
488	D₄	—	134	2	no	—
489	D₄	—	134	2	no	—
490	D₄	—	134	1	no	—
491	C₂×C₄	—	2	2	no	—
492	C₂×C₄	—	2	4	no	—
493	C₂×C₄	—	2	1	no	—
494	C₂×C₄	—	2	1	no	—
495	C₂×C₄	—	2	1	no	—
496	C₂×C₄	—	4	1	no	—
497	C₂×C₄	—	4	2	no	—
498	C₂×C₄	—	4	2	no	—
499	C₂×C₄	—	4	2	no	—
500	C₂×C₄	—	4	1	no	—
501	C₂×C₄	—	4	1	no	—
502	C₂×C₄	—	8	1	no	—
503	C₂×C₄	—	268	2	no	—
504	C₂×C₄	—	536	1	no	—
505	C₂×C₄	—	134	1	no	—
506	C₂×C₄	—	134	1	no	—
507	C₂×C₄	—	134	1	no	—
508	C₂×C₄	—	134	2	no	—
509	C₂×C₄	—	134	2	no	—
510	C₂×C₄	(C₂²⋊D₆₇)⋊C₄	1	1	yes	characteristic
511	C₂×C₄	(C₂²⋊D₆₇)⋊C₄	1	1	yes	characteristic
512	C₂×C₄	(C₂²⋊D₆₇)⋊C₄	1	1	yes	characteristic
513	C₂×C₄	(C₂²⋊D₆₇)⋊C₄	1	1	yes	characteristic
514	C₂×C₄	C₂²×C₄×D₆₇	1	1	yes	characteristic
515	C₂×C₄	C₂×D₄×D₆₇	1	1	yes	characteristic
516	C₂×C₄	#41×D₆₇	1	1	yes	characteristic
517	C₂×C₄	#41×D₆₇	1	1	yes	characteristic
518	C₂×C₄	#14524088	1	1	yes	characteristic
519	C₂×C₄	#14524091	1	1	yes	characteristic
520	C₈	—	4	2	no	—
521	C₈	—	268	2	no	—
522	C₈	—	134	2	no	—
523	C₈	—	134	2	no	—
524	C₈	—	134	1	no	—
525	C₈	—	134	1	no	—
526	C₈	—	134	1	no	—
527	C₈	—	134	1	no	—
528	C₂²	—	2	1	no	—
529	C₂²	—	2	1	no	—
530	C₂²	—	4	1	no	—
531	C₂²	—	4	1	no	—
532	C₂²	—	4	1	no	—
533	C₂²	—	268	1	no	—
534	C₂²	—	268	1	no	—
535	C₂²	—	134	2	no	—
536	C₂²	C₂×(C₂²⋊D₆₇)⋊C₄	1	1	yes	characteristic
537	C₂²	#16864080	1	1	yes	characteristic
538	C₂²	#16864093	1	1	yes	characteristic
539	C₄	—	2	1	no	—
540	C₄	—	2	1	no	—
541	C₄	—	2	1	no	—
542	C₄	—	4	4	no	—
543	C₄	—	4	1	no	—
544	C₄	—	4	1	no	—
545	C₄	—	4	1	no	—
546	C₄	—	268	1	no	—
547	C₄	—	268	1	no	—
548	C₄	—	134	2	no	—
549	C₄	#16863510	1	2	yes	—
550	C₄	#117×D₆₇	1	1	yes	characteristic
551	C₄	#16863877	1	1	yes	characteristic
552	C₄	#16864043	1	1	yes	characteristic
553	C₄	#16864056	1	1	yes	characteristic
554	C₂	—	2	1	no	—
555	C₂	—	2	1	no	—
556	C₂	—	8	1	no	—
557	C₂	—	268	2	no	—
558	C₂	#22563131	1	1	yes	characteristic
559	C₂	C₂×#16863510	1	1	yes	characteristic
560	C₂	#22569122	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×D₄×D₆₇
commutator	C₂×C₄×C₆₇	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₆₇
Fitting	C₆₇×#1806	C₂
socle	C₂²×C₆₇	C₂×C₄×D₄
prime radical	C₂×#14524063	C₂²
cosocle	C₂²×C₄×C₆₇	C₂⁴
hypercenter	#1806	D₆₇
hypocenter	C₆₇	#22783
LCS	G ▷ C₂×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1806

Complements

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

N	H ≅ G/N	type
2. #22568875	C₂	semidirect
3. #22568778	C₂	semidirect
4. #22560910	C₂	semidirect
5. #22560806	C₂	semidirect
7. #22560346	C₂	semidirect
8. #22560230	C₂	semidirect
9. C₄⋊C₄×C₆₇⋊C₈	C₂	semidirect
10. #22560043	C₂	semidirect
11. C₆₇×#1806	C₂	semidirect
15. C₄×#11761618	C₄	semidirect
16. C₄×#11761618	C₄	semidirect
17. C₄×#11761615	C₄	semidirect
18. C₄×#11761615	C₄	semidirect
19. C₄×#11761616	C₄	semidirect
20. C₄×#11761616	C₄	semidirect
21. C₄×#11761613	C₄	semidirect
22. C₄×#11761613	C₄	semidirect
289. #1806	D₆₇	semidirect
292. C₆₇	#22783	semidirect

Maximal subgroups

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

subgroup	index
C₄×#14524063	2
#22568875	2
#22568778	2
#22560910	2
#22560806	2
#22560533	2
#22560346	2
#22560230	2
C₄⋊C₄×C₆₇⋊C₈	2
#22560043	2
C₆₇×#1806	2
#22783	67