Subgroups — #31010490

Order 17152.

534 conjugacy classes of subgroups.

The subgroup lattice has 318 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×Dic₂₆₈	C₂	1	1	yes	maximal, characteristic
2	#22562370	C₂	1	4	yes	maximal, complemented
3	#22562358	C₂	1	2	yes	maximal, complemented
4	#22562348	C₂	1	2	yes	maximal, complemented
5	#22561998	C₂	1	2	yes	maximal
6	#22561965	C₂	1	1	yes	maximal, characteristic
7	#22561928	C₂	1	2	yes	maximal
8	C₆₇×#3160	C₂	1	1	yes	maximal, characteristic
9	C₂²×Dic₂₆₈	C₂²	1	1	yes	characteristic
10	C₂×C₄×Dic₁₃₄	—	2	1	no	—
11	C₂×#14523983	C₂²	1	2	yes	—
12	C₄×Dic₂₆₈	—	2	1	no	—
13	C₄×Dic₂₆₈	—	2	1	no	—
14	C₄×Dic₂₆₈	C₂²	1	2	yes	—
15	C₄×Dic₂₆₈	C₂²	1	2	yes	—
16	C₂×#14523977	C₂²	1	1	yes	characteristic
17	C₂×#14523968	—	2	1	no	—
18	#16863223	C₂²	1	4	yes	—
19	#16863221	C₂²	1	4	yes	—
20	C₄⋊(C₆₇⋊C₁₆)	C₂²	1	2	yes	—
21	C₄⋊(C₆₇⋊C₁₆)	C₂²	1	2	yes	—
22	#16863199	C₂²	1	4	yes	—
23	#16863199	C₂²	1	4	yes	—
24	#533×C₆₇	—	2	1	no	—
25	#500×C₆₇	C₂²	1	1	yes	characteristic
26	#500×C₆₇	C₂²	1	1	yes	characteristic
27	#499×C₆₇	—	2	1	no	—
28	#499×C₆₇	C₂²	1	2	yes	—
29	#498×C₆₇	—	2	1	no	—
30	#427×C₆₇	—	2	1	no	—
31	#427×C₆₇	C₂²	1	2	yes	—
32	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
33	C₂²×Dic₁₃₄	—	2	1	no	—
34	C₂×Dic₂₆₈	—	2	1	no	—
35	C₂×Dic₂₆₈	—	2	2	no	—
36	C₂×Dic₂₆₈	Q₈	1	4	yes	complemented
37	C₂×Dic₂₆₈	C₂³	1	2	yes	—
38	C₄×Dic₁₃₄	—	2	2	no	—
39	C₄×Dic₁₃₄	—	2	2	no	—
40	C₄×Dic₁₃₄	—	4	1	no	—
41	C₄×Dic₁₃₄	—	4	1	no	—
42	C₂×C₄⋊Dic₆₇	—	2	1	no	—
43	C₂×Dic₆₇⋊C₄	—	4	1	no	—
44	C₂×C₄×Dic₆₇	—	4	1	no	—
45	#14523983	—	2	4	no	—
46	C₂×C₆₇⋊C₁₆	C₂³	1	4	yes	—
47	#14523977	—	2	1	no	—
48	#14523977	C₂³	1	2	yes	—
49	#14523968	—	2	2	no	—
50	#14523968	—	2	2	no	—
51	C₂²×C₈×C₆₇	D₄	1	1	yes	characteristic
52	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
53	C₄⋊Q₈×C₆₇	—	2	1	no	—
54	C₄⋊Q₈×C₆₇	—	2	1	no	—
55	#125×C₆₇	—	2	2	no	—
56	C₂²⋊Q₈×C₆₇	—	4	1	no	—
57	C₂²⋊Q₈×C₆₇	—	4	1	no	—
58	C₄×Q₈×C₆₇	—	4	1	no	—
59	C₄×Q₈×C₆₇	—	4	1	no	—
60	#117×C₆₇	—	2	2	no	—
61	C₂×C₄²×C₆₇	D₄	1	1	yes	characteristic
62	#107×C₆₇	—	2	1	no	—
63	#107×C₆₇	—	2	1	no	—
64	#107×C₆₇	C₂³	1	2	yes	—
65	#107×C₆₇	C₂³	1	2	yes	—
66	#106×C₆₇	—	2	2	no	—
67	#106×C₆₇	—	2	2	no	—
68	C₄×C₈×C₆₇	D₄	1	1	yes	characteristic
69	C₄×C₈×C₆₇	D₄	1	1	yes	characteristic
70	C₄×C₈×C₆₇	C₂³	1	1	yes	characteristic
71	C₄×C₈×C₆₇	C₂³	1	1	yes	characteristic
72	C₂×Dic₁₃₄	—	2	4	no	—
73	C₂×Dic₁₃₄	—	2	2	no	—
74	C₂×Dic₁₃₄	—	4	1	no	—
75	C₂×Dic₁₃₄	—	4	2	no	—
76	C₂²×Dic₆₇	—	4	1	no	—
77	C₂²×Dic₆₇	—	4	1	no	—
78	Dic₂₆₈	—	2	4	no	—
79	Dic₂₆₈	—	2	2	no	—
80	Dic₂₆₈	C₂×Q₈	1	4	yes	—
81	C₄⋊Dic₆₇	—	2	2	no	—
82	C₄⋊Dic₆₇	—	4	1	no	—
83	Dic₆₇⋊C₄	—	4	2	no	—
84	Dic₆₇⋊C₄	—	4	2	no	—
85	C₄×Dic₆₇	—	4	2	no	—
86	C₄×Dic₆₇	—	4	2	no	—
87	C₆₇⋊C₁₆	—	2	4	no	—
88	C₂×Q₈×C₆₇	—	4	1	no	—
89	C₂×Q₈×C₆₇	—	4	1	no	—
90	C₂²×C₄×C₆₇	D₈	1	1	yes	characteristic
91	C₂²×C₄×C₆₇	D₈	1	1	yes	characteristic
92	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
93	C₂×C₈×C₆₇	—	2	2	no	—
94	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
95	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
96	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
97	C₂×C₈×C₆₇	C₂×Q₈	1	2	yes	—
98	C₂×C₈×C₆₇	#41	1	2	yes	—
99	C₂×C₈×C₆₇	C₂⁴	1	1	yes	characteristic
100	C₄⋊C₄×C₆₇	—	2	2	no	—
101	C₄⋊C₄×C₆₇	—	2	2	no	—
102	C₄⋊C₄×C₆₇	—	4	2	no	—
103	C₄⋊C₄×C₆₇	—	4	2	no	—
104	C₄⋊C₄×C₆₇	—	4	1	no	—
105	C₄⋊C₄×C₆₇	—	4	1	no	—
106	C₂²⋊C₄×C₆₇	—	4	2	no	—
107	C₄²×C₆₇	—	4	2	no	—
108	C₄²×C₆₇	D₈	1	1	yes	characteristic
109	C₄²×C₆₇	D₈	1	1	yes	characteristic
110	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
111	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
112	C₂³×C₆₇	C₂×D₈	1	1	yes	characteristic
113	Q₈×C₆₇	—	8	2	no	—
114	Q₈×C₆₇	—	8	2	no	—
115	C₂×C₄×C₆₇	—	2	2	no	—
116	C₂×C₄×C₆₇	—	4	1	no	—
117	C₂×C₄×C₆₇	—	4	2	no	—
118	C₂×C₄×C₆₇	—	4	2	no	—
119	C₂×C₄×C₆₇	C₂²⋊Q₈	1	2	yes	—
120	C₂×C₄×C₆₇	C₂×D₈	1	1	yes	characteristic
121	C₂×C₄×C₆₇	C₂×D₈	1	1	yes	characteristic
122	C₂×C₄×C₆₇	C₂×D₈	1	1	yes	characteristic
123	C₂×C₄×C₆₇	C₂×D₈	1	1	yes	characteristic
124	C₂×C₄×C₆₇	C₂×D₈	1	1	yes	characteristic
125	C₂×C₄×C₆₇	#137	1	2	yes	—
126	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
127	C₂×Dic₆₇	—	4	2	no	—
128	C₂×Dic₆₇	—	4	4	no	—
129	C₂×Dic₆₇	—	4	2	no	—
130	C₂×Dic₆₇	—	8	2	no	—
131	Dic₁₃₄	—	2	4	no	—
132	Dic₁₃₄	—	4	2	no	—
133	Dic₁₃₄	—	4	4	no	—
134	C₈×C₆₇	—	2	1	no	—
135	C₈×C₆₇	—	2	1	no	—
136	C₈×C₆₇	C₂²⋊Q₈	1	2	yes	—
137	C₈×C₆₇	C₂²×Q₈	1	1	yes	characteristic
138	C₈×C₆₇	C₂×#41	1	1	yes	characteristic
139	C₂²×C₆₇	—	2	1	no	—
140	C₂²×C₆₇	—	2	1	no	—
141	C₂²×C₆₇	#474	1	2	yes	—
142	C₂²×C₆₇	C₂²×D₈	1	1	yes	characteristic
143	C₄×C₆₇	—	2	1	no	—
144	C₄×C₆₇	—	2	1	no	—
145	C₄×C₆₇	—	2	1	no	—
146	C₄×C₆₇	—	8	2	no	—
147	C₄×C₆₇	—	8	2	no	—
148	C₄×C₆₇	#474	1	2	yes	—
149	C₄×C₆₇	#510	1	2	yes	—
150	C₄×C₆₇	C₂×C₂²⋊Q₈	1	1	yes	characteristic
151	C₄×C₆₇	C₂×#137	1	1	yes	characteristic
152	Dic₆₇	—	4	4	no	—
153	Dic₆₇	—	8	2	no	—
154	#33628	—	67	1	no	maximal, Sylow
155	C₂×C₆₇	—	2	2	no	—
156	C₂×C₆₇	C₂×#474	1	1	yes	characteristic
157	C₂×C₆₇	C₂×#510	1	1	yes	characteristic
158	C₂×C₆₇	#3414	1	1	yes	characteristic
159	#3160	D₆₇	1	1	yes	characteristic
160	C₂×C₄×Q₁₆	—	67	1	no	—
161	#2229	—	67	4	no	—
162	#2227	—	67	2	no	—
163	#2225	—	67	2	no	—
164	#2159	—	67	2	no	—
165	#2150	—	67	1	no	—
166	#2141	—	67	2	no	—
167	C₆₇	#33628	1	1	yes	characteristic, complemented, Sylow
168	C₂²×Q₁₆	—	67	1	no	—
169	#533	—	2	1	no	—
170	C₂×C₄×Q₈	—	134	1	no	—
171	C₂×M₃₂	—	67	2	no	—
172	#500	C₂×D₆₇	1	1	yes	characteristic
173	#500	C₂×D₆₇	1	1	yes	characteristic
174	#499	—	2	1	no	—
175	#499	C₂×D₆₇	1	2	yes	—
176	#498	—	2	1	no	—
177	C₄×Q₁₆	—	67	2	no	—
178	C₄×Q₁₆	—	67	2	no	—
179	C₄×Q₁₆	—	134	1	no	—
180	C₄×Q₁₆	—	134	1	no	—
181	#427	—	2	1	no	—
182	#427	C₂×D₆₇	1	2	yes	—
183	C₂×#107	—	67	1	no	—
184	C₂×#103	—	134	1	no	—
185	C₂×C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
186	#367	—	67	4	no	—
187	#366	—	67	4	no	—
188	C₄⋊C₁₆	—	67	2	no	—
189	C₄⋊C₁₆	—	67	2	no	—
190	#358	—	67	4	no	—
191	#358	—	67	4	no	—
192	C₂²×Q₈	—	134	1	no	—
193	C₂×Q₁₆	—	67	4	no	—
194	C₂×Q₁₆	—	67	2	no	—
195	C₂×Q₁₆	—	134	1	no	—
196	C₂×Q₁₆	—	134	2	no	—
197	C₂²×C₈	C₂²⋊D₆₇	1	1	yes	characteristic
198	C₂²×C₈	C₂²×D₆₇	1	1	yes	characteristic
199	C₄⋊Q₈	—	2	1	no	—
200	C₄⋊Q₈	—	2	1	no	—
201	#125	—	2	2	no	—
202	C₂²⋊Q₈	—	4	1	no	—
203	C₂²⋊Q₈	—	4	1	no	—
204	C₄×Q₈	—	4	1	no	—
205	C₄×Q₈	—	4	1	no	—
206	C₄×Q₈	—	268	1	no	—
207	C₄×Q₈	—	268	1	no	—
208	C₄×Q₈	—	134	2	no	—
209	C₄×Q₈	—	134	2	no	—
210	#117	—	2	2	no	—
211	C₂×C₄⋊C₄	—	268	1	no	—
212	C₂×C₄⋊C₄	—	134	1	no	—
213	C₂×C₄²	—	268	1	no	—
214	C₂×C₄²	C₂²⋊D₆₇	1	1	yes	characteristic
215	M₃₂	—	134	4	no	—
216	C₂×C₁₆	—	67	4	no	—
217	#107	—	2	1	no	—
218	#107	—	2	1	no	—
219	#107	—	67	2	no	—
220	#107	—	134	1	no	—
221	#107	C₂²×D₆₇	1	2	yes	—
222	#107	C₂²×D₆₇	1	2	yes	—
223	#106	—	2	2	no	—
224	#106	—	2	2	no	—
225	#103	—	134	2	no	—
226	#103	—	134	2	no	—
227	C₄×C₈	C₂²⋊D₆₇	1	1	yes	characteristic
228	C₄×C₈	C₂²⋊D₆₇	1	1	yes	characteristic
229	C₄×C₈	C₂²×D₆₇	1	1	yes	characteristic
230	C₄×C₈	C₂²×D₆₇	1	1	yes	characteristic
231	C₂×Q₈	—	4	1	no	—
232	C₂×Q₈	—	4	1	no	—
233	C₂×Q₈	—	268	1	no	—
234	C₂×Q₈	—	268	2	no	—
235	C₂×Q₈	—	134	4	no	—
236	C₂×Q₈	—	134	2	no	—
237	C₂²×C₄	—	268	1	no	—
238	C₂²×C₄	—	268	1	no	—
239	C₂²×C₄	#11761613	1	1	yes	characteristic
240	C₂²×C₄	#11761613	1	1	yes	characteristic
241	C₂²×C₄	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
242	Q₁₆	—	67	4	no	—
243	Q₁₆	—	134	4	no	—
244	Q₁₆	—	134	2	no	—
245	C₂×C₈	—	2	2	no	—
246	C₂×C₈	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
247	C₂×C₈	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
248	C₂×C₈	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
249	C₂×C₈	Q₈×D₆₇	1	2	yes	—
250	C₂×C₈	#11761629	1	2	yes	—
251	C₂×C₈	C₂³×D₆₇	1	1	yes	characteristic
252	C₄⋊C₄	—	2	2	no	—
253	C₄⋊C₄	—	2	2	no	—
254	C₄⋊C₄	—	4	2	no	—
255	C₄⋊C₄	—	4	2	no	—
256	C₄⋊C₄	—	4	1	no	—
257	C₄⋊C₄	—	4	1	no	—
258	C₄⋊C₄	—	268	2	no	—
259	C₄⋊C₄	—	268	2	no	—
260	C₄⋊C₄	—	268	1	no	—
261	C₄⋊C₄	—	134	2	no	—
262	C₂²⋊C₄	—	4	2	no	—
263	C₄²	—	4	2	no	—
264	C₄²	—	268	2	no	—
265	C₄²	—	268	2	no	—
266	C₄²	#11761613	1	1	yes	characteristic
267	C₄²	#11761613	1	1	yes	characteristic
268	C₄²	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
269	C₄²	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
270	C₁₆	—	134	4	no	—
271	C₂³	C₂×#11761613	1	1	yes	characteristic
272	Q₈	—	8	2	no	—
273	Q₈	—	8	2	no	—
274	Q₈	—	268	4	no	—
275	Q₈	—	268	2	no	—
276	Q₈	—	134	4	no	—
277	C₂×C₄	—	2	2	no	—
278	C₂×C₄	—	4	1	no	—
279	C₂×C₄	—	4	2	no	—
280	C₂×C₄	—	4	2	no	—
281	C₂×C₄	—	268	2	no	—
282	C₂×C₄	—	268	4	no	—
283	C₂×C₄	—	268	2	no	—
284	C₂×C₄	—	536	2	no	—
285	C₂×C₄	#14524019	1	2	yes	—
286	C₂×C₄	C₂×#11761613	1	1	yes	characteristic
287	C₂×C₄	C₂×#11761613	1	1	yes	characteristic
288	C₂×C₄	C₂×#11761613	1	1	yes	characteristic
289	C₂×C₄	C₂×#11761613	1	1	yes	characteristic
290	C₂×C₄	C₂×#11761613	1	1	yes	characteristic
291	C₂×C₄	#14524074	1	2	yes	—
292	C₂×C₄	C₂²×C₂²⋊D₆₇	1	1	yes	characteristic
293	C₈	—	2	1	no	—
294	C₈	—	2	1	no	—
295	C₈	#14524019	1	2	yes	—
296	C₈	C₂×Q₈×D₆₇	1	1	yes	characteristic
297	C₈	C₂×#11761629	1	1	yes	characteristic
298	C₂²	—	2	1	no	—
299	C₂²	—	2	1	no	—
300	C₂²	#16863673	1	2	yes	—
301	C₂²	C₂²×#11761613	1	1	yes	characteristic
302	C₄	—	2	1	no	—
303	C₄	—	2	1	no	—
304	C₄	—	2	1	no	—
305	C₄	—	8	2	no	—
306	C₄	—	8	2	no	—
307	C₄	—	268	4	no	—
308	C₄	—	536	2	no	—
309	C₄	#16863673	1	2	yes	—
310	C₄	#16863836	1	2	yes	—
311	C₄	C₂×#14524019	1	1	yes	characteristic
312	C₄	C₂×#14524074	1	1	yes	characteristic
313	C₂	—	2	2	no	—
314	C₂	C₂×#16863673	1	1	yes	characteristic
315	C₂	C₂×#16863836	1	1	yes	characteristic
316	C₂	#22573343	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂²×#11761613
commutator, cosocle	C₂×C₈×C₆₇	C₂⁴
Frattini	C₂×C₈	C₂³×D₆₇
Fitting	C₆₇×#3160	C₂
socle	C₂²×C₆₇	C₂²×D₈
prime radical	C₂³×C₆₇	C₂×D₈
hypercenter	#3160	D₆₇
hypocenter	C₆₇	#33628
LCS	G ▷ C₂×C₈×C₆₇ ▷ C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ #3160

Complements

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

N	H ≅ G/N	type
2. #22562370	C₂	semidirect
3. #22562358	C₂	semidirect
4. #22562348	C₂	semidirect
36. C₂×Dic₂₆₈	Q₈	semidirect
167. C₆₇	#33628	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄×Dic₂₆₈	2
#22562370	2
#22562358	2
#22562348	2
#22561998	2
#22561965	2
#22561928	2
C₆₇×#3160	2
#33628	67