Subgroups — #30764258

Order 17152.

622 conjugacy classes of subgroups.

The subgroup lattice has 296 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22568799	C₂	1	1	yes	maximal, characteristic
2	#22559947	C₂	1	2	yes	maximal
3	C₂×#16863156	C₂	1	2	yes	maximal
4	C₂×C₆₇⋊#338	C₂	1	1	yes	maximal, characteristic
5	C₆₇×#1728	C₂	1	1	yes	maximal, characteristic
6	C₂²×#11761612	C₂²	1	2	yes	—
7	C₂×#14524002	C₂²	1	2	yes	—
8	#16863438	—	2	4	no	—
9	#16863434	—	2	2	no	—
10	#16863164	—	2	4	no	—
11	#16863156	—	2	4	no	—
12	C₆₇⋊#338	C₄	1	4	yes	complemented
13	C₂²×M₁₆×C₆₇	C₂²	1	1	yes	characteristic
14	C₂²×M₁₆×C₆₇	C₂²	1	1	yes	characteristic
15	C₂²×C₄²×C₆₇	C₂²	1	1	yes	characteristic
16	#336×C₆₇	—	2	2	no	—
17	C₂×#11761612	—	2	2	no	—
18	C₂×#11761612	—	2	2	no	—
19	C₂×#11761612	—	2	2	no	—
20	C₂×#11761612	—	2	2	no	—
21	C₂×#11761612	—	2	2	no	—
22	C₂×#11761612	C₂×C₄	1	4	yes	—
23	C₂²×C₆₇⋊C₈	—	2	2	no	—
24	#14524002	—	2	2	no	—
25	#14524002	—	2	2	no	—
26	#14524002	—	2	2	no	—
27	#14524002	—	2	2	no	—
28	C₂×C₄⋊Dic₆₇	D₄	1	4	yes	—
29	C₂×C₂²⋊Dic₆₇	—	2	2	no	—
30	C₂×C₄×Dic₆₇	—	2	2	no	—
31	#14523977	—	2	4	no	—
32	#14523975	—	2	4	no	—
33	C₂³×C₄×C₆₇	—	2	1	no	—
34	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
35	C₂×M₁₆×C₆₇	—	2	2	no	—
36	C₂×M₁₆×C₆₇	—	2	2	no	—
37	C₂×M₁₆×C₆₇	—	2	1	no	—
38	C₂×M₁₆×C₆₇	—	2	2	no	—
39	C₂×M₁₆×C₆₇	—	2	2	no	—
40	C₂×M₁₆×C₆₇	C₂×C₄	1	2	yes	—
41	C₂×M₁₆×C₆₇	Q₈	1	2	yes	—
42	C₂×M₁₆×C₆₇	Q₈	1	2	yes	—
43	C₂²×C₈×C₆₇	—	2	1	no	—
44	C₂²×C₈×C₆₇	D₄	1	2	yes	—
45	C₂×C₄²×C₆₇	—	2	2	no	—
46	C₂×C₄²×C₆₇	—	2	2	no	—
47	C₂×C₄²×C₆₇	—	2	1	no	—
48	C₂×C₄²×C₆₇	D₄	1	2	yes	—
49	C₂²×Dic₆₇	—	2	4	no	—
50	#11761612	—	2	4	no	—
51	#11761612	—	2	4	no	—
52	#11761612	—	4	2	no	—
53	#11761612	—	4	2	no	—
54	C₂×C₆₇⋊C₈	—	2	4	no	—
55	C₂×C₆₇⋊C₈	—	4	2	no	—
56	C₂×C₆₇⋊C₈	—	4	2	no	—
57	C₄⋊Dic₆₇	—	2	4	no	—
58	C₄⋊Dic₆₇	—	2	4	no	—
59	C₂²⋊Dic₆₇	—	4	2	no	—
60	C₂²⋊Dic₆₇	—	4	2	no	—
61	C₄×Dic₆₇	—	4	2	no	—
62	C₄×Dic₆₇	—	4	2	no	—
63	C₂⁴×C₆₇	#41	1	1	yes	characteristic
64	C₂²×C₄×C₆₇	—	2	2	no	—
65	C₂²×C₄×C₆₇	—	4	4	no	—
66	C₂²×C₄×C₆₇	—	4	2	no	—
67	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
68	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
69	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
70	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
71	C₂²×C₄×C₆₇	C₂×Q₈	1	1	yes	characteristic
72	C₂²×C₄×C₆₇	C₂×Q₈	1	1	yes	characteristic
73	C₂²×C₄×C₆₇	#41	1	2	yes	—
74	C₂²×C₄×C₆₇	#41	1	2	yes	—
75	C₂²×C₄×C₆₇	#41	1	1	yes	characteristic
76	C₂²×C₄×C₆₇	#41	1	2	yes	—
77	M₁₆×C₆₇	—	2	4	no	—
78	M₁₆×C₆₇	—	2	4	no	—
79	M₁₆×C₆₇	—	2	4	no	—
80	M₁₆×C₆₇	—	4	2	no	—
81	C₂×C₈×C₆₇	—	2	2	no	—
82	C₂×C₈×C₆₇	—	2	2	no	—
83	C₂×C₈×C₆₇	—	2	2	no	—
84	C₂×C₈×C₆₇	—	4	1	no	—
85	C₂×C₈×C₆₇	—	4	1	no	—
86	C₂×C₈×C₆₇	C₄⋊C₄	1	4	yes	—
87	C₄²×C₆₇	—	2	4	no	—
88	C₄²×C₆₇	—	4	1	no	—
89	C₄²×C₆₇	—	4	1	no	—
90	C₂³×C₆₇	—	4	2	no	—
91	C₂³×C₆₇	C₄×Q₈	1	1	yes	characteristic
92	C₂³×C₆₇	#121	1	1	yes	characteristic
93	C₂³×C₆₇	#123	1	2	yes	—
94	C₂³×C₆₇	#125	1	1	yes	characteristic
95	C₂³×C₆₇	#126	1	2	yes	—
96	C₂×C₄×C₆₇	—	2	4	no	—
97	C₂×C₄×C₆₇	—	4	2	no	—
98	C₂×C₄×C₆₇	—	4	8	no	—
99	C₂×C₄×C₆₇	—	4	2	no	—
100	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
101	C₂×C₄×C₆₇	#117	1	2	yes	—
102	C₂×C₄×C₆₇	C₄×D₄	1	2	yes	—
103	C₂×C₄×C₆₇	C₄×D₄	1	1	yes	characteristic
104	C₂×C₄×C₆₇	#120	1	1	yes	characteristic
105	C₂×C₄×C₆₇	#121	1	2	yes	—
106	C₂×C₄×C₆₇	C₂²⋊Q₈	1	2	yes	—
107	C₂×C₄×C₆₇	C₂²⋊Q₈	1	2	yes	—
108	C₂×C₄×C₆₇	C₂²⋊Q₈	1	1	yes	characteristic
109	C₂×C₄×C₆₇	#123	1	1	yes	characteristic
110	C₂×C₄×C₆₇	#123	1	2	yes	—
111	C₂×C₄×C₆₇	#124	1	2	yes	—
112	C₂×C₄×C₆₇	#124	1	1	yes	characteristic
113	C₂×C₄×C₆₇	#124	1	1	yes	characteristic
114	C₂×C₄×C₆₇	#125	1	2	yes	—
115	C₂×C₄×C₆₇	#125	1	2	yes	—
116	C₂×C₄×C₆₇	#126	1	2	yes	—
117	C₂×C₄×C₆₇	C₄⋊Q₈	1	1	yes	characteristic
118	C₂×Dic₆₇	—	4	4	no	—
119	C₂×Dic₆₇	—	8	4	no	—
120	C₈×C₆₇	—	2	4	no	—
121	C₈×C₆₇	—	4	2	no	—
122	C₆₇⋊C₈	—	4	4	no	—
123	C₂²×C₆₇	—	2	4	no	—
124	C₂²×C₆₇	—	2	2	no	—
125	C₂²×C₆₇	—	4	2	no	—
126	C₂²×C₆₇	—	4	1	no	—
127	C₂²×C₆₇	—	4	1	no	—
128	C₂²×C₆₇	#387	1	2	yes	—
129	C₂²×C₆₇	#389	1	1	yes	characteristic
130	C₂²×C₆₇	#394	1	1	yes	characteristic
131	C₂²×C₆₇	#397	1	1	yes	characteristic
132	C₂²×C₆₇	#400	1	2	yes	—
133	C₄×C₆₇	—	4	4	no	—
134	C₄×C₆₇	#382	1	1	yes	characteristic
135	C₄×C₆₇	#385	1	1	yes	characteristic
136	C₄×C₆₇	#388	1	2	yes	—
137	C₄×C₆₇	#393	1	1	yes	characteristic
138	C₄×C₆₇	#398	1	2	yes	—
139	C₄×C₆₇	#400	1	1	yes	characteristic
140	Dic₆₇	—	8	4	no	—
141	#11294	—	67	1	no	maximal, Sylow
142	C₂×C₆₇	—	2	1	no	—
143	C₂×C₆₇	—	2	2	no	—
144	C₂×C₆₇	—	2	1	no	—
145	C₂×C₆₇	—	2	2	no	—
146	C₂×C₆₇	#1442	1	1	yes	characteristic
147	C₂×C₆₇	#2050	1	2	yes	—
148	#2911	—	67	1	no	—
149	#1742	—	67	2	no	—
150	C₂×#339	—	67	2	no	—
151	C₂×#338	—	67	1	no	—
152	#1728	D₆₇	1	1	yes	characteristic
153	C₆₇	#11294	1	1	yes	characteristic, complemented, Sylow
154	C₂²×M₁₆	—	67	2	no	—
155	C₂²×M₁₆	C₂×D₆₇	1	1	yes	characteristic
156	C₂²×M₁₆	C₂×D₆₇	1	1	yes	characteristic
157	C₂×#117	—	67	2	no	—
158	C₂²×C₄²	C₂×D₆₇	1	1	yes	characteristic
159	#428	—	134	4	no	—
160	#427	—	134	2	no	—
161	#343	—	134	4	no	—
162	#339	—	134	4	no	—
163	#338	—	67	4	no	—
164	#336	—	2	2	no	—
165	C₂³×C₄	—	2	1	no	—
166	C₂³×C₄	C₂²×D₆₇	1	1	yes	characteristic
167	C₂×M₁₆	—	2	2	no	—
168	C₂×M₁₆	—	2	2	no	—
169	C₂×M₁₆	—	2	1	no	—
170	C₂×M₁₆	—	2	2	no	—
171	C₂×M₁₆	—	2	2	no	—
172	C₂×M₁₆	—	67	4	no	—
173	C₂×M₁₆	—	134	2	no	—
174	C₂×M₁₆	—	134	2	no	—
175	C₂×M₁₆	—	134	2	no	—
176	C₂×M₁₆	—	134	2	no	—
177	C₂×M₁₆	—	134	2	no	—
178	C₂×M₁₆	Dic₁₃₄	1	2	yes	—
179	C₂×M₁₆	Dic₁₃₄	1	2	yes	—
180	C₂×M₁₆	C₄×D₆₇	1	2	yes	—
181	C₂²×C₈	—	2	1	no	—
182	C₂²×C₈	—	134	2	no	—
183	C₂²×C₈	C₂²⋊D₆₇	1	2	yes	—
184	#117	—	134	2	no	—
185	#117	—	134	2	no	—
186	#117	—	134	2	no	—
187	#117	—	134	2	no	—
188	C₂×C₄⋊C₄	—	67	4	no	—
189	C₂×C₂²⋊C₄	—	134	2	no	—
190	C₂×C₄²	—	2	2	no	—
191	C₂×C₄²	—	2	2	no	—
192	C₂×C₄²	—	2	1	no	—
193	C₂×C₄²	—	134	2	no	—
194	C₂×C₄²	D₂₆₈	1	2	yes	—
195	#107	—	134	4	no	—
196	#106	—	134	4	no	—
197	C₂⁴	#11761628	1	1	yes	characteristic
198	C₂²×C₄	—	2	2	no	—
199	C₂²×C₄	—	4	4	no	—
200	C₂²×C₄	—	4	2	no	—
201	C₂²×C₄	—	134	4	no	—
202	C₂²×C₄	C₂×C₄×D₆₇	1	1	yes	characteristic
203	C₂²×C₄	D₄×D₆₇	1	2	yes	—
204	C₂²×C₄	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
205	C₂²×C₄	C₂×D₂₆₈	1	1	yes	characteristic
206	C₂²×C₄	C₂×Dic₁₃₄	1	1	yes	characteristic
207	C₂²×C₄	C₂×Dic₁₃₄	1	1	yes	characteristic
208	C₂²×C₄	#11761627	1	2	yes	—
209	C₂²×C₄	#11761627	1	2	yes	—
210	C₂²×C₄	#11761628	1	1	yes	characteristic
211	C₂²×C₄	#11761629	1	2	yes	—
212	M₁₆	—	2	4	no	—
213	M₁₆	—	2	4	no	—
214	M₁₆	—	2	4	no	—
215	M₁₆	—	4	2	no	—
216	M₁₆	—	268	2	no	—
217	M₁₆	—	268	2	no	—
218	M₁₆	—	134	4	no	—
219	M₁₆	—	134	4	no	—
220	C₂×C₈	—	2	2	no	—
221	C₂×C₈	—	2	2	no	—
222	C₂×C₈	—	2	2	no	—
223	C₂×C₈	—	4	1	no	—
224	C₂×C₈	—	4	1	no	—
225	C₂×C₈	—	268	2	no	—
226	C₂×C₈	—	268	2	no	—
227	C₂×C₈	—	134	4	no	—
228	C₂×C₈	Dic₆₇⋊C₄	1	4	yes	—
229	C₄⋊C₄	—	134	4	no	—
230	C₄⋊C₄	—	134	4	no	—
231	C₂²⋊C₄	—	268	2	no	—
232	C₂²⋊C₄	—	268	2	no	—
233	C₄²	—	2	4	no	—
234	C₄²	—	4	1	no	—
235	C₄²	—	4	1	no	—
236	C₄²	—	268	2	no	—
237	C₄²	—	268	2	no	—
238	C₂³	—	4	2	no	—
239	C₂³	C₄×Dic₁₃₄	1	1	yes	characteristic
240	C₂³	#14524017	1	1	yes	characteristic
241	C₂³	#14524028	1	2	yes	—
242	C₂³	#14524035	1	1	yes	characteristic
243	C₂³	#14524036	1	2	yes	—
244	C₂×C₄	—	2	4	no	—
245	C₂×C₄	—	4	2	no	—
246	C₂×C₄	—	4	8	no	—
247	C₂×C₄	—	4	2	no	—
248	C₂×C₄	—	268	4	no	—
249	C₂×C₄	—	536	4	no	—
250	C₂×C₄	C₂×Dic₆₇⋊C₄	1	1	yes	characteristic
251	C₂×C₄	#14524001	1	2	yes	—
252	C₂×C₄	(C₂²⋊D₆₇)⋊C₄	1	2	yes	—
253	C₂×C₄	C₄×D₂₆₈	1	1	yes	characteristic
254	C₂×C₄	#14524011	1	1	yes	characteristic
255	C₂×C₄	#14524018	1	2	yes	—
256	C₂×C₄	#14524020	1	2	yes	—
257	C₂×C₄	#14524020	1	2	yes	—
258	C₂×C₄	#14524022	1	1	yes	characteristic
259	C₂×C₄	#14524024	1	1	yes	characteristic
260	C₂×C₄	#14524027	1	2	yes	—
261	C₂×C₄	#14524029	1	2	yes	—
262	C₂×C₄	#14524030	1	1	yes	characteristic
263	C₂×C₄	#14524032	1	1	yes	characteristic
264	C₂×C₄	#14524034	1	2	yes	—
265	C₂×C₄	#14524034	1	2	yes	—
266	C₂×C₄	#14524038	1	2	yes	—
267	C₂×C₄	#14524043	1	1	yes	characteristic
268	C₈	—	2	4	no	—
269	C₈	—	4	2	no	—
270	C₈	—	268	4	no	—
271	C₂²	—	2	2	no	—
272	C₂²	—	2	4	no	—
273	C₂²	—	4	2	no	—
274	C₂²	—	4	1	no	—
275	C₂²	—	4	1	no	—
276	C₂²	#16863283	1	2	yes	—
277	C₂²	#16863294	1	1	yes	characteristic
278	C₂²	#16863313	1	1	yes	characteristic
279	C₂²	#16863321	1	1	yes	characteristic
280	C₂²	#16863338	1	2	yes	—
281	C₄	—	4	4	no	—
282	C₄	—	536	4	no	—
283	C₄	#16863261	1	1	yes	characteristic
284	C₄	#16863268	1	1	yes	characteristic
285	C₄	#16863288	1	2	yes	—
286	C₄	#16863308	1	1	yes	characteristic
287	C₄	#16863328	1	2	yes	—
288	C₄	#16863335	1	1	yes	characteristic
289	C₂	—	2	1	no	—
290	C₂	—	2	2	no	—
291	C₂	—	2	1	no	—
292	C₂	—	2	2	no	—
293	C₂	#22558465	1	1	yes	characteristic
294	C₂	#22561496	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	#14524032
commutator	C₂²×C₄×C₆₇	C₂²×C₄
Frattini	C₂³×C₄	C₂²×D₆₇
Fitting	C₆₇×#1728	C₂
socle	C₂²×C₆₇	#394
prime radical	C₂⁴×C₆₇	#41
cosocle	C₂³×C₄×C₆₇	C₂³
hypercenter	#1728	D₆₇
hypocenter	C₆₇	#11294
LCS	G ▷ C₂²×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ #1728

Complements

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

N	H ≅ G/N	type
12. C₆₇⋊#338	C₄	semidirect
153. C₆₇	#11294	semidirect

Maximal subgroups

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

subgroup	index
#22568799	2
#22559947	2
C₂×#16863156	2
C₂×C₆₇⋊#338	2
C₆₇×#1728	2
#11294	67