Subgroups — #31010794

Order 17152.

546 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	#22570795	C₂	1	1	yes	maximal, characteristic
2	#22562411	C₂	1	2	yes	maximal
3	#22562361	C₂	1	2	yes	maximal, complemented
4	#22562259	C₂	1	2	yes	maximal, complemented
5	#22562143	C₂	1	2	yes	maximal
6	C₄×#14523989	C₂	1	4	yes	maximal, complemented
7	#22562008	C₂	1	1	yes	maximal, characteristic
8	C₂×C₄×Q₁₆×C₆₇	C₂	1	1	yes	maximal, characteristic
9	#16863953	—	2	1	no	—
10	C₂×#14523989	C₂²	1	4	yes	—
11	#16863801	C₂²	1	1	yes	characteristic
12	#16863776	C₂²	1	1	yes	characteristic
13	#16863747	—	2	1	no	—
14	#16863637	—	2	1	no	—
15	#16863637	C₂²	1	2	yes	—
16	C₄×Dic₂₆₈	—	2	1	no	—
17	C₄×Dic₂₆₈	C₂²	1	2	yes	—
18	#16863221	C₂²	1	4	yes	—
19	C₄⋊(C₆₇⋊C₁₆)	C₂²	1	2	yes	—
20	#16863200	C₂²	1	4	yes	—
21	#16863199	C₂²	1	4	yes	—
22	C₂²⋊(C₆₇⋊C₁₆)	C₂²	1	2	yes	—
23	C₄×C₆₇⋊C₁₆	C₂²	1	2	yes	—
24	C₂²×Q₁₆×C₆₇	C₂²	1	1	yes	characteristic
25	C₂×C₄×Q₈×C₆₇	—	2	1	no	—
26	C₄×Q₁₆×C₆₇	—	2	1	no	—
27	C₄×Q₁₆×C₆₇	—	2	1	no	—
28	C₄×Q₁₆×C₆₇	C₂²	1	2	yes	—
29	C₄×Q₁₆×C₆₇	C₂²	1	2	yes	—
30	C₂×#107×C₆₇	C₂²	1	1	yes	characteristic
31	C₂×#103×C₆₇	—	2	1	no	—
32	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
33	C₂×Dic₂₆₈	—	2	1	no	—
34	C₂×Dic₂₆₈	C₂³	1	2	yes	—
35	#14524043	—	2	1	no	—
36	#14524043	—	2	1	no	—
37	#14524035	—	4	1	no	—
38	#14524022	—	2	2	no	—
39	#14524022	—	4	1	no	—
40	C₄×Dic₁₃₄	—	2	2	no	—
41	C₄×Dic₁₃₄	—	4	1	no	—
42	#14524002	—	4	1	no	—
43	#14523989	—	2	8	no	—
44	C₂×C₆₇⋊C₁₆	C₂³	1	4	yes	—
45	#14523977	—	2	1	no	—
46	#14523977	C₂³	1	2	yes	—
47	#14523968	—	2	2	no	—
48	#14523968	—	2	2	no	—
49	C₂²×Q₈×C₆₇	—	2	1	no	—
50	C₂×Q₁₆×C₆₇	—	2	4	no	—
51	C₂×Q₁₆×C₆₇	—	2	1	no	—
52	C₂×Q₁₆×C₆₇	C₂³	1	2	yes	—
53	C₂²×C₈×C₆₇	D₄	1	1	yes	characteristic
54	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
55	C₄×Q₈×C₆₇	—	2	2	no	—
56	C₄×Q₈×C₆₇	—	2	2	no	—
57	C₄×Q₈×C₆₇	—	4	1	no	—
58	C₄×Q₈×C₆₇	—	4	1	no	—
59	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
60	C₂×C₄⋊C₄×C₆₇	—	4	1	no	—
61	C₂×C₄²×C₆₇	—	4	1	no	—
62	C₂×C₄²×C₆₇	D₄	1	1	yes	characteristic
63	#107×C₆₇	—	2	1	no	—
64	#107×C₆₇	C₂³	1	2	yes	—
65	#103×C₆₇	—	2	2	no	—
66	#103×C₆₇	—	2	2	no	—
67	C₄×C₈×C₆₇	D₄	1	1	yes	characteristic
68	C₄×C₈×C₆₇	D₄	1	1	yes	characteristic
69	C₄×C₈×C₆₇	C₂³	1	1	yes	characteristic
70	C₄×C₈×C₆₇	C₂³	1	1	yes	characteristic
71	C₂×Dic₁₃₄	—	2	2	no	—
72	C₂×Dic₁₃₄	—	4	1	no	—
73	Dic₂₆₈	—	2	4	no	—
74	Dic₂₆₈	—	4	2	no	—
75	C₄⋊Dic₆₇	—	2	2	no	—
76	C₄⋊Dic₆₇	—	4	1	no	—
77	C₄⋊Dic₆₇	—	4	1	no	—
78	C₄⋊Dic₆₇	—	4	1	no	—
79	Dic₆₇⋊C₄	—	4	2	no	—
80	Dic₆₇⋊C₄	—	4	2	no	—
81	C₂²⋊Dic₆₇	—	4	2	no	—
82	C₄×Dic₆₇	—	4	2	no	—
83	C₆₇⋊C₁₆	—	2	4	no	—
84	C₂×Q₈×C₆₇	—	2	4	no	—
85	C₂×Q₈×C₆₇	—	2	2	no	—
86	C₂×Q₈×C₆₇	—	4	2	no	—
87	C₂×Q₈×C₆₇	—	4	1	no	—
88	C₂²×C₄×C₆₇	—	4	1	no	—
89	C₂²×C₄×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	Q₁₆×C₆₇	—	2	4	no	—
94	Q₁₆×C₆₇	—	2	2	no	—
95	Q₁₆×C₆₇	#41	1	4	yes	—
96	C₂×C₈×C₆₇	—	2	2	no	—
97	C₂×C₈×C₆₇	—	4	1	no	—
98	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
99	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
100	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
101	C₂×C₈×C₆₇	C₂⁴	1	1	yes	characteristic
102	C₄⋊C₄×C₆₇	—	2	2	no	—
103	C₄⋊C₄×C₆₇	—	4	1	no	—
104	C₄⋊C₄×C₆₇	—	4	2	no	—
105	C₄⋊C₄×C₆₇	—	4	2	no	—
106	C₄²×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	—	2	4	no	—
114	Q₈×C₆₇	—	4	4	no	—
115	Q₈×C₆₇	—	4	2	no	—
116	C₂×C₄×C₆₇	—	2	1	no	—
117	C₂×C₄×C₆₇	—	2	1	no	—
118	C₂×C₄×C₆₇	—	4	1	no	—
119	C₂×C₄×C₆₇	—	4	4	no	—
120	C₂×C₄×C₆₇	—	4	2	no	—
121	C₂×C₄×C₆₇	—	4	2	no	—
122	C₂×C₄×C₆₇	—	8	2	no	—
123	C₂×C₄×C₆₇	Q₃₂	1	4	yes	complemented
124	C₂×C₄×C₆₇	C₂×D₈	1	1	yes	characteristic
125	C₂×C₄×C₆₇	C₂×D₈	1	1	yes	characteristic
126	C₂×C₄×C₆₇	C₂×D₈	1	1	yes	characteristic
127	C₂×C₄×C₆₇	C₂×D₈	1	1	yes	characteristic
128	C₂×C₄×C₆₇	C₂×D₈	1	1	yes	characteristic
129	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
130	C₂×Dic₆₇	—	4	2	no	—
131	C₂×Dic₆₇	—	4	2	no	—
132	Dic₁₃₄	—	4	2	no	—
133	Dic₁₃₄	—	4	2	no	—
134	Dic₁₃₄	—	8	2	no	—
135	C₈×C₆₇	—	2	1	no	—
136	C₈×C₆₇	—	2	1	no	—
137	C₈×C₆₇	—	2	1	no	—
138	C₈×C₆₇	C₂×#41	1	2	yes	—
139	C₂²×C₆₇	—	2	1	no	—
140	C₂²×C₆₇	—	2	1	no	—
141	C₂²×C₆₇	C₂×Q₃₂	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	—	4	4	no	—
147	C₄×C₆₇	—	8	2	no	—
148	C₄×C₆₇	C₂×Q₃₂	1	2	yes	—
149	C₄×C₆₇	C₂×Q₃₂	1	2	yes	—
150	C₄×C₆₇	#525	1	1	yes	characteristic
151	C₄×C₆₇	#575	1	1	yes	characteristic
152	Dic₆₇	—	8	2	no	—
153	Dic₆₇	—	8	2	no	—
154	#33664	—	67	1	no	maximal, Sylow
155	C₂×C₆₇	—	2	2	no	—
156	C₂×C₆₇	#3100	1	1	yes	characteristic
157	C₂×C₆₇	C₂²×Q₃₂	1	1	yes	characteristic
158	C₂×C₆₇	#3414	1	1	yes	characteristic
159	#3170	—	67	1	no	—
160	C₂×C₄×Q₁₆	D₆₇	1	1	yes	characteristic
161	#2236	—	67	2	no	—
162	#2227	—	67	2	no	—
163	#2209	—	67	2	no	—
164	#2188	—	67	2	no	—
165	C₄×Q₃₂	—	67	4	no	—
166	#2162	—	67	1	no	—
167	C₆₇	#33664	1	1	yes	characteristic, complemented, Sylow
168	C₂²×Q₁₆	C₂×D₆₇	1	1	yes	characteristic
169	#533	—	134	1	no	—
170	C₂×C₄×Q₈	—	2	1	no	—
171	C₂×Q₃₂	—	67	4	no	—
172	#500	—	67	1	no	—
173	#494	—	67	1	no	—
174	#487	—	134	1	no	—
175	#467	—	67	2	no	—
176	#467	—	134	1	no	—
177	C₄×Q₁₆	—	2	1	no	—
178	C₄×Q₁₆	—	2	1	no	—
179	C₄×Q₁₆	—	67	2	no	—
180	C₄×Q₁₆	—	134	1	no	—
181	C₄×Q₁₆	C₂×D₆₇	1	2	yes	—
182	C₄×Q₁₆	C₂×D₆₇	1	2	yes	—
183	C₂×#107	C₂×D₆₇	1	1	yes	characteristic
184	C₂×#103	—	2	1	no	—
185	C₂×C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
186	#366	—	67	4	no	—
187	C₄⋊C₁₆	—	67	2	no	—
188	#358	—	67	4	no	—
189	#358	—	67	4	no	—
190	C₂²⋊C₁₆	—	67	2	no	—
191	C₄×C₁₆	—	67	2	no	—
192	C₂²×Q₈	—	2	1	no	—
193	C₂×Q₁₆	—	2	4	no	—
194	C₂×Q₁₆	—	2	1	no	—
195	C₂×Q₁₆	—	67	2	no	—
196	C₂×Q₁₆	—	134	1	no	—
197	C₂×Q₁₆	C₂²×D₆₇	1	2	yes	—
198	C₂²×C₈	C₂²⋊D₆₇	1	1	yes	characteristic
199	C₂²×C₈	C₂²×D₆₇	1	1	yes	characteristic
200	C₄⋊Q₈	—	134	1	no	—
201	C₄⋊Q₈	—	134	1	no	—
202	#125	—	268	1	no	—
203	C₂²⋊Q₈	—	268	1	no	—
204	C₂²⋊Q₈	—	134	2	no	—
205	C₄×Q₈	—	2	2	no	—
206	C₄×Q₈	—	2	2	no	—
207	C₄×Q₈	—	4	1	no	—
208	C₄×Q₈	—	4	1	no	—
209	C₄×Q₈	—	268	1	no	—
210	C₄×Q₈	—	134	2	no	—
211	#117	—	268	1	no	—
212	C₂×C₄⋊C₄	—	2	1	no	—
213	C₂×C₄⋊C₄	—	4	1	no	—
214	C₂×C₄²	—	4	1	no	—
215	C₂×C₄²	C₂²⋊D₆₇	1	1	yes	characteristic
216	Q₃₂	—	134	8	no	—
217	C₂×C₁₆	—	67	4	no	—
218	#107	—	2	1	no	—
219	#107	—	67	2	no	—
220	#107	—	134	1	no	—
221	#107	C₂²×D₆₇	1	2	yes	—
222	#103	—	2	2	no	—
223	#103	—	2	2	no	—
224	#103	—	134	2	no	—
225	#103	—	134	2	no	—
226	C₄×C₈	C₂²⋊D₆₇	1	1	yes	characteristic
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	C₂×Q₈	—	2	4	no	—
231	C₂×Q₈	—	2	2	no	—
232	C₂×Q₈	—	4	2	no	—
233	C₂×Q₈	—	4	1	no	—
234	C₂×Q₈	—	268	1	no	—
235	C₂×Q₈	—	134	2	no	—
236	C₂²×C₄	—	4	1	no	—
237	C₂²×C₄	—	4	1	no	—
238	C₂²×C₄	#11761613	1	1	yes	characteristic
239	C₂²×C₄	#11761613	1	1	yes	characteristic
240	C₂²×C₄	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
241	Q₁₆	—	2	4	no	—
242	Q₁₆	—	2	2	no	—
243	Q₁₆	—	268	2	no	—
244	Q₁₆	—	134	4	no	—
245	Q₁₆	#11761628	1	4	yes	—
246	C₂×C₈	—	2	2	no	—
247	C₂×C₈	—	4	1	no	—
248	C₂×C₈	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
249	C₂×C₈	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
250	C₂×C₈	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
251	C₂×C₈	C₂³×D₆₇	1	1	yes	characteristic
252	C₄⋊C₄	—	2	2	no	—
253	C₄⋊C₄	—	4	1	no	—
254	C₄⋊C₄	—	4	2	no	—
255	C₄⋊C₄	—	4	2	no	—
256	C₄⋊C₄	—	268	1	no	—
257	C₄⋊C₄	—	268	1	no	—
258	C₄⋊C₄	—	268	1	no	—
259	C₄⋊C₄	—	268	2	no	—
260	C₄⋊C₄	—	268	2	no	—
261	C₄⋊C₄	—	134	2	no	—
262	C₂²⋊C₄	—	268	2	no	—
263	C₄²	—	4	2	no	—
264	C₄²	—	4	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	—	2	4	no	—
273	Q₈	—	4	4	no	—
274	Q₈	—	4	2	no	—
275	Q₈	—	268	2	no	—
276	Q₈	—	268	2	no	—
277	Q₈	—	536	2	no	—
278	C₂×C₄	—	2	1	no	—
279	C₂×C₄	—	2	1	no	—
280	C₂×C₄	—	4	1	no	—
281	C₂×C₄	—	4	4	no	—
282	C₂×C₄	—	4	2	no	—
283	C₂×C₄	—	4	2	no	—
284	C₂×C₄	—	8	2	no	—
285	C₂×C₄	—	268	2	no	—
286	C₂×C₄	—	268	2	no	—
287	C₂×C₄	#14523989	1	4	yes	complemented
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	C₂×#11761613	1	1	yes	characteristic
292	C₂×C₄	C₂×#11761613	1	1	yes	characteristic
293	C₂×C₄	C₂²×C₂²⋊D₆₇	1	1	yes	characteristic
294	C₈	—	2	1	no	—
295	C₈	—	2	1	no	—
296	C₈	—	2	1	no	—
297	C₈	C₂×#11761628	1	2	yes	—
298	C₂²	—	2	1	no	—
299	C₂²	—	2	1	no	—
300	C₂²	C₂×#14523989	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	—	4	4	no	—
306	C₄	—	8	2	no	—
307	C₄	—	536	2	no	—
308	C₄	—	536	2	no	—
309	C₄	C₂×#14523989	1	2	yes	—
310	C₄	C₂×#14523989	1	2	yes	—
311	C₄	#16863907	1	1	yes	characteristic
312	C₄	#16864194	1	1	yes	characteristic
313	C₂	—	2	2	no	—
314	C₂	#22570209	1	1	yes	characteristic
315	C₂	C₂²×#14523989	1	1	yes	characteristic
316	C₂	#22573343	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×#11761613
commutator, cosocle	C₂×C₈×C₆₇	C₂⁴
Frattini	C₂×C₈	C₂³×D₆₇
Fitting	C₂×C₄×Q₁₆×C₆₇	C₂
socle	C₂²×C₆₇	C₂²×D₈
prime radical	C₂³×C₆₇	C₂×D₈
hypercenter	C₂×C₄×Q₁₆	D₆₇
hypocenter	C₆₇	#33664
LCS	G ▷ C₂×C₈×C₆₇ ▷ C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×C₄×Q₁₆

Complements

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

N	H ≅ G/N	type
3. #22562361	C₂	semidirect
4. #22562259	C₂	semidirect
6. C₄×#14523989	C₂	semidirect
123. C₂×C₄×C₆₇	Q₃₂	semidirect
167. C₆₇	#33664	semidirect
287. C₂×C₄	#14523989	semidirect

Maximal subgroups

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

subgroup	index
#22570795	2
#22562411	2
#22562361	2
#22562259	2
#22562143	2
C₄×#14523989	2
#22562008	2
C₂×C₄×Q₁₆×C₆₇	2
#33664	67