Subgroups — #30853115

Order 17152.

542 conjugacy classes of subgroups.

The subgroup lattice has 312 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22573014	C₂	1	2	yes	maximal
2	#22573010	C₂	1	2	yes	maximal
3	#22559819	C₂	1	1	yes	maximal, characteristic
4	#22559764	C₂	1	2	yes	maximal
5	#22559763	C₂	1	2	yes	maximal
6	#22559454	C₂	1	2	yes	maximal
7	#22559453	C₂	1	2	yes	maximal
8	#22559107	C₂	1	1	yes	maximal, characteristic
9	C₆₇×#1591	C₂	1	1	yes	maximal, characteristic
10	#16864125	C₂²	1	2	yes	—
11	#16864124	C₂²	1	2	yes	—
12	#16863801	C₂²	1	1	yes	characteristic
13	#16863801	C₂²	1	1	yes	characteristic
14	#16863765	—	2	2	no	—
15	#16863763	—	2	2	no	—
16	#16863747	C₂²	1	2	yes	—
17	#16863747	C₂²	1	2	yes	—
18	#16863704	—	2	2	no	—
19	#16863702	—	2	2	no	—
20	#16863682	C₂²	1	2	yes	—
21	#16863681	C₂²	1	2	yes	—
22	#16863607	—	2	2	no	—
23	#16863607	C₂²	1	2	yes	—
24	#16863606	C₂²	1	2	yes	—
25	#16863604	—	2	2	no	—
26	#16863526	C₂²	1	2	yes	—
27	#16863504	—	2	2	no	—
28	#16863500	—	2	2	no	—
29	C₄×Dic₂₆₈	C₂²	1	1	yes	characteristic
30	#16863149	C₂²	1	1	yes	characteristic
31	#16863143	C₂²	1	2	yes	—
32	#16863141	C₂²	1	2	yes	—
33	#16863128	C₂²	1	2	yes	—
34	#446×C₆₇	C₂²	1	2	yes	—
35	C₄×Q₁₆×C₆₇	C₂²	1	1	yes	characteristic
36	#335×C₆₇	C₂²	1	1	yes	characteristic
37	Q₈⋊C₈×C₆₇	C₂²	1	2	yes	—
38	C₈²×C₆₇	C₂²	1	1	yes	characteristic
39	C₂×Dic₂₆₈	—	2	2	no	—
40	C₂×Dic₂₆₈	D₄	1	1	yes	characteristic
41	C₂×#11761618	—	2	2	no	—
42	#14524043	—	2	2	no	—
43	#14524043	C₂³	1	2	yes	—
44	#14524043	C₂³	1	2	yes	—
45	#14524041	—	2	2	no	—
46	#14524035	—	2	2	no	—
47	#14524034	—	2	2	no	—
48	#14524034	—	4	2	no	—
49	#14524033	—	4	2	no	—
50	Q₈×Dic₆₇	—	2	2	no	—
51	C₄×Dic₁₃₄	—	2	2	no	—
52	C₄×Dic₁₃₄	—	2	2	no	—
53	C₄×Dic₁₃₄	C₂³	1	2	yes	—
54	#14524008	—	2	2	no	—
55	#14523977	—	4	1	no	—
56	#14523977	—	4	1	no	—
57	#14523977	D₄	1	1	yes	characteristic
58	#14523975	—	2	2	no	—
59	#14523974	—	2	2	no	—
60	C₄⋊(C₆₇⋊C₈)	C₂³	1	2	yes	—
61	Dic₆₇⋊C₈	—	2	2	no	—
62	#14523968	—	2	2	no	—
63	#14523968	—	2	2	no	—
64	#14523968	—	2	2	no	—
65	#14523968	—	2	2	no	—
66	#14523968	—	2	2	no	—
67	#14523968	—	2	2	no	—
68	#14523968	D₄	1	2	yes	—
69	#14523967	—	2	2	no	—
70	#14523966	—	2	2	no	—
71	#14523953	—	2	2	no	—
72	C₂×Q₁₆×C₆₇	D₄	1	1	yes	characteristic
73	C₄×Q₈×C₆₇	C₂³	1	2	yes	—
74	#107×C₆₇	D₄	1	1	yes	characteristic
75	C₄⋊C₈×C₆₇	—	2	2	no	—
76	C₄⋊C₈×C₆₇	C₂³	1	2	yes	—
77	#103×C₆₇	D₄	1	2	yes	—
78	#97×C₆₇	—	2	2	no	—
79	C₄×C₈×C₆₇	C₂³	1	1	yes	characteristic
80	C₄×C₈×C₆₇	C₂³	1	1	yes	characteristic
81	C₄×C₈×C₆₇	C₂³	1	1	yes	characteristic
82	C₂×Dic₁₃₄	—	2	2	no	—
83	C₂×Dic₁₃₄	—	2	2	no	—
84	C₂×Dic₁₃₄	C₂×D₄	1	2	yes	—
85	Dic₂₆₈	—	4	1	no	—
86	Dic₂₆₈	—	4	4	no	—
87	#11761618	—	4	4	no	—
88	C₂×C₆₇⋊C₈	—	2	2	no	—
89	C₄⋊Dic₆₇	—	2	2	no	—
90	C₄⋊Dic₆₇	—	2	2	no	—
91	C₄⋊Dic₆₇	—	2	2	no	—
92	C₄⋊Dic₆₇	—	2	2	no	—
93	C₄⋊Dic₆₇	—	4	2	no	—
94	C₄⋊Dic₆₇	—	4	2	no	—
95	C₄⋊Dic₆₇	C₂×D₄	1	2	yes	—
96	Dic₆₇⋊C₄	—	2	2	no	—
97	Dic₆₇⋊C₄	—	4	2	no	—
98	Dic₆₇⋊C₄	—	4	2	no	—
99	Dic₆₇⋊C₄	—	4	2	no	—
100	Dic₆₇⋊C₄	—	4	2	no	—
101	C₄×Dic₆₇	—	2	2	no	—
102	C₄×Dic₆₇	—	4	2	no	—
103	C₄×Dic₆₇	—	4	2	no	—
104	C₂×Q₈×C₆₇	C₂×D₄	1	2	yes	—
105	Q₁₆×C₆₇	—	4	1	no	—
106	C₂×C₈×C₆₇	—	2	1	no	—
107	C₂×C₈×C₆₇	—	2	2	no	—
108	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
109	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
110	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
111	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
112	C₄⋊C₄×C₆₇	—	2	2	no	—
113	C₄⋊C₄×C₆₇	C₂×D₄	1	2	yes	—
114	C₄²×C₆₇	—	2	2	no	—
115	C₄²×C₆₇	C₂⁴	1	1	yes	characteristic
116	Q₈×C₆₇	—	2	2	no	—
117	Q₈×C₆₇	—	2	2	no	—
118	C₂×C₄×C₆₇	—	2	2	no	—
119	C₂×C₄×C₆₇	—	2	2	no	—
120	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
121	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
122	C₂×C₄×C₆₇	H(5,𝔽₂)	1	1	yes	characteristic
123	C₂×Dic₆₇	—	2	2	no	—
124	C₂×Dic₆₇	—	2	2	no	—
125	C₂×Dic₆₇	—	4	2	no	—
126	C₂×Dic₆₇	—	4	2	no	—
127	Dic₁₃₄	—	2	2	no	—
128	Dic₁₃₄	—	2	2	no	—
129	Dic₁₃₄	—	4	2	no	—
130	Dic₁₃₄	—	4	2	no	—
131	Dic₁₃₄	—	4	2	no	—
132	Dic₁₃₄	—	4	2	no	—
133	C₈×C₆₇	—	2	2	no	—
134	C₈×C₆₇	—	2	1	no	—
135	C₈×C₆₇	—	2	1	no	—
136	C₈×C₆₇	—	2	1	no	—
137	C₈×C₆₇	—	2	1	no	—
138	C₈×C₆₇	—	4	2	no	—
139	C₆₇⋊C₈	—	4	2	no	—
140	C₂²×C₆₇	D₄²	1	1	yes	characteristic
141	C₄×C₆₇	—	2	1	no	—
142	C₄×C₆₇	—	4	2	no	—
143	C₄×C₆₇	—	4	2	no	—
144	C₄×C₆₇	#575	1	1	yes	characteristic
145	C₄×C₆₇	#575	1	1	yes	characteristic
146	C₄×C₆₇	#578	1	1	yes	characteristic
147	C₄×C₆₇	#578	1	1	yes	characteristic
148	Dic₆₇	—	4	2	no	—
149	Dic₆₇	—	4	2	no	—
150	Dic₆₇	—	8	2	no	—
151	Dic₆₇	—	8	2	no	—
152	#20321	—	67	1	no	maximal, Sylow
153	C₂×C₆₇	#3277	1	1	yes	characteristic
154	C₂×C₆₇	#3277	1	1	yes	characteristic
155	C₂×C₆₇	#3293	1	1	yes	characteristic
156	#3374	—	67	2	no	—
157	#3374	—	67	2	no	—
158	#1710	—	67	1	no	—
159	#1700	—	67	2	no	—
160	#1700	—	67	2	no	—
161	#1648	—	67	2	no	—
162	#1648	—	67	2	no	—
163	#1591	—	67	1	no	—
164	#1591	D₆₇	1	1	yes	characteristic
165	C₆₇	#20321	1	1	yes	characteristic, complemented, Sylow
166	#557	—	67	2	no	—
167	#557	—	67	2	no	—
168	#500	—	67	1	no	—
169	#500	—	67	1	no	—
170	#491	—	134	2	no	—
171	#491	—	134	2	no	—
172	#487	—	67	2	no	—
173	#487	—	67	2	no	—
174	#479	—	134	2	no	—
175	#479	—	134	2	no	—
176	#475	—	67	2	no	—
177	#475	—	67	2	no	—
178	#462	—	67	2	no	—
179	#462	—	67	2	no	—
180	#462	—	134	2	no	—
181	#462	—	134	2	no	—
182	#446	—	67	2	no	—
183	#446	C₂×D₆₇	1	2	yes	—
184	#441	—	134	2	no	—
185	#441	—	134	2	no	—
186	C₄×Q₁₆	—	67	1	no	—
187	C₄×Q₁₆	C₂×D₆₇	1	1	yes	characteristic
188	#335	—	67	1	no	—
189	#335	C₂×D₆₇	1	1	yes	characteristic
190	#332	—	67	2	no	—
191	#332	—	67	2	no	—
192	Q₈⋊C₈	—	67	2	no	—
193	Q₈⋊C₈	C₂×D₆₇	1	2	yes	—
194	C₈²	C₂×D₆₇	1	1	yes	characteristic
195	C₂×Q₁₆	—	67	1	no	—
196	C₂×Q₁₆	—	134	2	no	—
197	C₂×Q₁₆	—	134	2	no	—
198	C₂×Q₁₆	D₂₆₈	1	1	yes	characteristic
199	C₄⋊Q₈	—	67	2	no	—
200	C₄⋊Q₈	—	67	2	no	—
201	C₄⋊Q₈	—	134	2	no	—
202	C₄⋊Q₈	—	134	2	no	—
203	#125	—	268	2	no	—
204	#125	—	268	2	no	—
205	#125	—	134	2	no	—
206	#125	—	134	2	no	—
207	C₄×Q₈	—	67	2	no	—
208	C₄×Q₈	—	134	2	no	—
209	C₄×Q₈	—	134	2	no	—
210	C₄×Q₈	—	134	2	no	—
211	C₄×Q₈	—	134	2	no	—
212	C₄×Q₈	C₂²×D₆₇	1	2	yes	—
213	#107	—	268	1	no	—
214	#107	—	268	1	no	—
215	#107	—	67	1	no	—
216	#107	D₂₆₈	1	1	yes	characteristic
217	#106	—	134	2	no	—
218	#106	—	134	2	no	—
219	C₄⋊C₈	—	2	2	no	—
220	C₄⋊C₈	—	67	2	no	—
221	C₄⋊C₈	—	134	2	no	—
222	C₄⋊C₈	C₂²×D₆₇	1	2	yes	—
223	#103	—	67	2	no	—
224	#103	—	134	2	no	—
225	#103	—	134	2	no	—
226	#103	—	134	2	no	—
227	#103	—	134	2	no	—
228	#103	—	134	2	no	—
229	#103	—	134	2	no	—
230	#103	—	134	2	no	—
231	#103	—	134	2	no	—
232	#103	D₂₆₈	1	2	yes	—
233	#97	—	2	2	no	—
234	#97	—	134	2	no	—
235	C₄×C₈	C₂²×D₆₇	1	1	yes	characteristic
236	C₄×C₈	C₂²×D₆₇	1	1	yes	characteristic
237	C₄×C₈	C₂²×D₆₇	1	1	yes	characteristic
238	C₂×Q₈	—	67	2	no	—
239	C₂×Q₈	—	134	2	no	—
240	C₂×Q₈	—	134	2	no	—
241	C₂×Q₈	C₂×D₂₆₈	1	2	yes	—
242	Q₁₆	—	4	1	no	—
243	Q₁₆	—	268	1	no	—
244	Q₁₆	—	268	4	no	—
245	Q₁₆	—	268	4	no	—
246	C₂×C₈	—	2	1	no	—
247	C₂×C₈	—	2	2	no	—
248	C₂×C₈	—	134	2	no	—
249	C₂×C₈	D₄×D₆₇	1	1	yes	characteristic
250	C₂×C₈	D₄×D₆₇	1	1	yes	characteristic
251	C₂×C₈	C₂×D₂₆₈	1	1	yes	characteristic
252	C₂×C₈	C₂×D₂₆₈	1	1	yes	characteristic
253	C₄⋊C₄	—	2	2	no	—
254	C₄⋊C₄	—	268	2	no	—
255	C₄⋊C₄	—	268	2	no	—
256	C₄⋊C₄	—	268	2	no	—
257	C₄⋊C₄	—	268	2	no	—
258	C₄⋊C₄	—	268	2	no	—
259	C₄⋊C₄	—	268	2	no	—
260	C₄⋊C₄	—	67	2	no	—
261	C₄⋊C₄	—	134	2	no	—
262	C₄⋊C₄	—	134	2	no	—
263	C₄⋊C₄	—	134	2	no	—
264	C₄⋊C₄	—	134	2	no	—
265	C₄⋊C₄	—	134	2	no	—
266	C₄⋊C₄	C₂×D₂₆₈	1	2	yes	—
267	C₄²	—	2	2	no	—
268	C₄²	—	268	2	no	—
269	C₄²	—	268	2	no	—
270	C₄²	—	134	2	no	—
271	C₄²	C₂³×D₆₇	1	1	yes	characteristic
272	Q₈	—	2	2	no	—
273	Q₈	—	2	2	no	—
274	Q₈	—	268	2	no	—
275	Q₈	—	268	2	no	—
276	Q₈	—	268	2	no	—
277	Q₈	—	268	2	no	—
278	Q₈	—	134	2	no	—
279	Q₈	—	134	2	no	—
280	C₂×C₄	—	2	2	no	—
281	C₂×C₄	—	2	2	no	—
282	C₂×C₄	—	268	2	no	—
283	C₂×C₄	—	268	2	no	—
284	C₂×C₄	—	134	2	no	—
285	C₂×C₄	—	134	2	no	—
286	C₂×C₄	C₂×D₄×D₆₇	1	1	yes	characteristic
287	C₂×C₄	C₂²×D₂₆₈	1	1	yes	characteristic
288	C₂×C₄	#14524089	1	1	yes	characteristic
289	C₈	—	2	2	no	—
290	C₈	—	2	1	no	—
291	C₈	—	2	1	no	—
292	C₈	—	2	1	no	—
293	C₈	—	2	1	no	—
294	C₈	—	4	2	no	—
295	C₈	—	268	2	no	—
296	C₂²	D₄×D₂₆₈	1	1	yes	characteristic
297	C₄	—	2	1	no	—
298	C₄	—	4	2	no	—
299	C₄	—	4	2	no	—
300	C₄	—	268	2	no	—
301	C₄	—	268	2	no	—
302	C₄	—	536	2	no	—
303	C₄	—	536	2	no	—
304	C₄	#16864193	1	1	yes	characteristic
305	C₄	#16864196	1	1	yes	characteristic
306	C₄	#16864209	1	1	yes	characteristic
307	C₄	#16864213	1	1	yes	characteristic
308	C₂	#22571865	1	1	yes	characteristic
309	C₂	#22571867	1	1	yes	characteristic
310	C₂	#22572014	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	D₄×D₂₆₈
commutator, cosocle	C₄²×C₆₇	C₂⁴
Frattini	C₄²	C₂³×D₆₇
Fitting	C₆₇×#1591	C₂
socle, prime radical	C₂²×C₆₇	D₄²
hypercenter	#1591	D₆₇
hypocenter	C₆₇	#20321
LCS	G ▷ C₄²×C₆₇ ▷ C₂²×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #1591

Complements

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

N	H ≅ G/N	type
165. C₆₇	#20321	semidirect

Maximal subgroups

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

subgroup	index
#22573014	2
#22573010	2
#22559819	2
#22559764	2
#22559763	2
#22559454	2
#22559453	2
#22559107	2
C₆₇×#1591	2
#20321	67