Subgroups — #30764670

Order 17152.

634 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	C₂²×#14523975	C₂	1	1	yes	maximal, characteristic
2	C₂×#16863164	C₂	1	2	yes	maximal
3	#22559929	C₂	1	2	yes	maximal
4	#22559905	C₂	1	1	yes	maximal, characteristic
5	C₂×#341×C₆₇	C₂	1	1	yes	maximal, characteristic
6	C₂²×#11761612	C₂²	1	1	yes	characteristic
7	C₂×#14524002	C₂²	1	1	yes	characteristic
8	C₂²×C₄⋊Dic₆₇	C₂²	1	2	yes	—
9	C₂×#14523975	—	2	2	no	—
10	C₂×#14523975	—	2	2	no	—
11	C₂×#14523975	C₄	1	4	yes	complemented
12	#16863164	—	2	4	no	—
13	#16863159	—	2	4	no	—
14	#16863150	—	2	2	no	—
15	C₂²×M₁₆×C₆₇	C₂²	1	2	yes	—
16	C₂³×C₈×C₆₇	C₂²	1	1	yes	characteristic
17	#341×C₆₇	—	2	2	no	—
18	C₂³×Dic₆₇	—	2	2	no	—
19	C₂×#11761612	—	2	1	no	—
20	C₂×#11761612	—	2	2	no	—
21	C₂×#11761612	—	2	2	no	—
22	C₂×#11761612	D₄	1	2	yes	—
23	C₂²×C₆₇⋊C₈	—	2	1	no	—
24	#14524002	—	2	2	no	—
25	#14524002	—	2	2	no	—
26	C₂×C₄⋊Dic₆₇	—	2	2	no	—
27	C₂×C₄⋊Dic₆₇	—	2	2	no	—
28	C₂×C₄⋊Dic₆₇	—	2	2	no	—
29	C₂×C₄⋊Dic₆₇	—	2	2	no	—
30	C₂×C₄⋊Dic₆₇	—	2	2	no	—
31	C₂×C₄⋊Dic₆₇	C₂×C₄	1	4	yes	—
32	C₂×C₄⋊Dic₆₇	D₄	1	2	yes	—
33	C₂×C₂²⋊Dic₆₇	—	2	1	no	—
34	C₂×C₄×Dic₆₇	—	2	1	no	—
35	#14523975	—	2	8	no	—
36	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
37	C₂×M₁₆×C₆₇	—	2	2	no	—
38	C₂×M₁₆×C₆₇	—	2	2	no	—
39	C₂×M₁₆×C₆₇	—	2	2	no	—
40	C₂×M₁₆×C₆₇	—	2	2	no	—
41	C₂×M₁₆×C₆₇	—	2	2	no	—
42	C₂×M₁₆×C₆₇	Q₈	1	4	yes	—
43	C₂²×C₈×C₆₇	—	2	2	no	—
44	C₂²×C₈×C₆₇	—	2	2	no	—
45	C₂²×C₈×C₆₇	—	2	2	no	—
46	C₂²×C₈×C₆₇	C₂×C₄	1	2	yes	—
47	C₂²×C₈×C₆₇	D₄	1	2	yes	—
48	C₂²×C₈×C₆₇	D₄	1	2	yes	—
49	C₂²×Dic₆₇	—	2	2	no	—
50	C₂²×Dic₆₇	—	2	4	no	—
51	C₂²×Dic₆₇	—	4	2	no	—
52	C₂²×Dic₆₇	—	4	2	no	—
53	C₂²×Dic₆₇	—	8	4	no	—
54	#11761612	—	2	4	no	—
55	#11761612	—	4	2	no	—
56	C₂×C₆₇⋊C₈	—	2	2	no	—
57	C₂×C₆₇⋊C₈	—	4	1	no	—
58	C₂×C₆₇⋊C₈	—	4	1	no	—
59	C₄⋊Dic₆₇	—	2	4	no	—
60	C₄⋊Dic₆₇	—	2	4	no	—
61	C₄⋊Dic₆₇	—	2	4	no	—
62	C₄⋊Dic₆₇	—	4	2	no	—
63	C₄⋊Dic₆₇	—	4	2	no	—
64	C₂²⋊Dic₆₇	—	4	2	no	—
65	C₄×Dic₆₇	—	4	1	no	—
66	C₄×Dic₆₇	—	4	1	no	—
67	C₂⁴×C₆₇	#41	1	1	yes	characteristic
68	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
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×D₄	1	1	yes	characteristic
72	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
73	C₂²×C₄×C₆₇	C₂×Q₈	1	2	yes	—
74	C₂²×C₄×C₆₇	#41	1	2	yes	—
75	C₂²×C₄×C₆₇	#41	1	2	yes	—
76	C₂²×C₄×C₆₇	#41	1	2	yes	—
77	C₂²×C₄×C₆₇	#41	1	1	yes	characteristic
78	M₁₆×C₆₇	—	2	4	no	—
79	M₁₆×C₆₇	—	2	4	no	—
80	M₁₆×C₆₇	—	4	2	no	—
81	M₁₆×C₆₇	—	4	2	no	—
82	C₂×C₈×C₆₇	—	2	4	no	—
83	C₂×C₈×C₆₇	—	2	4	no	—
84	C₂×C₈×C₆₇	—	2	2	no	—
85	C₂×C₈×C₆₇	—	2	2	no	—
86	C₂×C₈×C₆₇	—	2	4	no	—
87	C₂×C₈×C₆₇	—	4	2	no	—
88	C₂×C₈×C₆₇	—	4	2	no	—
89	C₂×C₈×C₆₇	C₂²⋊C₄	1	4	yes	—
90	C₂³×C₆₇	—	2	4	no	—
91	C₂³×C₆₇	C₄×D₄	1	1	yes	characteristic
92	C₂³×C₆₇	#121	1	1	yes	characteristic
93	C₂³×C₆₇	#124	1	1	yes	characteristic
94	C₂³×C₆₇	#125	1	2	yes	—
95	C₂³×C₆₇	#126	1	2	yes	—
96	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
97	C₂×C₄×C₆₇	#117	1	2	yes	—
98	C₂×C₄×C₆₇	C₄×D₄	1	1	yes	characteristic
99	C₂×C₄×C₆₇	C₄×Q₈	1	2	yes	—
100	C₂×C₄×C₆₇	#121	1	1	yes	characteristic
101	C₂×C₄×C₆₇	C₂²⋊Q₈	1	2	yes	—
102	C₂×C₄×C₆₇	C₂²⋊Q₈	1	2	yes	—
103	C₂×C₄×C₆₇	C₂²⋊Q₈	1	2	yes	—
104	C₂×C₄×C₆₇	#123	1	2	yes	—
105	C₂×C₄×C₆₇	#123	1	2	yes	—
106	C₂×C₄×C₆₇	#123	1	2	yes	—
107	C₂×C₄×C₆₇	#123	1	1	yes	characteristic
108	C₂×C₄×C₆₇	#123	1	1	yes	characteristic
109	C₂×C₄×C₆₇	#124	1	1	yes	characteristic
110	C₂×C₄×C₆₇	#125	1	2	yes	—
111	C₂×C₄×C₆₇	#126	1	2	yes	—
112	C₂×C₄×C₆₇	#127	1	1	yes	characteristic
113	C₂×C₄×C₆₇	C₄⋊Q₈	1	1	yes	characteristic
114	C₂×Dic₆₇	—	4	4	no	—
115	C₂×Dic₆₇	—	4	2	no	—
116	C₂×Dic₆₇	—	4	4	no	—
117	C₂×Dic₆₇	—	8	4	no	—
118	C₂×Dic₆₇	—	8	4	no	—
119	C₂×Dic₆₇	—	8	4	no	—
120	C₈×C₆₇	—	2	4	no	—
121	C₈×C₆₇	—	4	4	no	—
122	C₆₇⋊C₈	—	4	2	no	—
123	C₂²×C₆₇	—	2	4	no	—
124	C₂²×C₆₇	—	2	4	no	—
125	C₂²×C₆₇	—	2	2	no	—
126	C₂²×C₆₇	—	2	2	no	—
127	C₂²×C₆₇	#387	1	2	yes	—
128	C₂²×C₆₇	#390	1	1	yes	characteristic
129	C₂²×C₆₇	#397	1	1	yes	characteristic
130	C₂²×C₆₇	#398	1	1	yes	characteristic
131	C₂²×C₆₇	#400	1	2	yes	—
132	C₂²×C₆₇	#471	1	4	yes	—
133	C₄×C₆₇	#381	1	1	yes	characteristic
134	C₄×C₆₇	#387	1	2	yes	—
135	C₄×C₆₇	#391	1	1	yes	characteristic
136	C₄×C₆₇	#397	1	1	yes	characteristic
137	C₄×C₆₇	#398	1	1	yes	characteristic
138	C₄×C₆₇	#399	1	2	yes	—
139	Dic₆₇	—	8	2	no	—
140	Dic₆₇	—	8	4	no	—
141	#11366	—	67	1	no	maximal, Sylow
142	C₂×C₆₇	—	2	2	no	—
143	C₂×C₆₇	—	2	2	no	—
144	C₂×C₆₇	#1445	1	1	yes	characteristic
145	C₂×C₆₇	#1977	1	2	yes	—
146	C₂×C₆₇	#2079	1	2	yes	—
147	C₂×C₆₇	#2090	1	2	yes	—
148	C₂²×#106	—	67	1	no	—
149	C₂×#343	—	67	2	no	—
150	C₂×#341	D₆₇	1	1	yes	characteristic
151	#1736	—	67	2	no	—
152	#1729	—	67	1	no	—
153	C₆₇	#11366	1	1	yes	characteristic, complemented, Sylow
154	C₂²×M₁₆	—	67	1	no	—
155	C₂²×M₁₆	C₂×D₆₇	1	2	yes	—
156	C₂³×C₈	C₂×D₆₇	1	1	yes	characteristic
157	C₂×#117	—	67	1	no	—
158	C₂²×C₄⋊C₄	—	67	2	no	—
159	C₂×#106	—	67	4	no	—
160	C₂×#106	—	134	2	no	—
161	C₂×#106	—	134	2	no	—
162	#343	—	134	4	no	—
163	#341	—	2	2	no	—
164	#340	—	134	4	no	—
165	#336	—	134	2	no	—
166	C₂³×C₄	—	134	2	no	—
167	C₂³×C₄	C₂²×D₆₇	1	1	yes	characteristic
168	C₂×M₁₆	—	2	2	no	—
169	C₂×M₁₆	—	2	2	no	—
170	C₂×M₁₆	—	2	2	no	—
171	C₂×M₁₆	—	2	2	no	—
172	C₂×M₁₆	—	2	2	no	—
173	C₂×M₁₆	—	67	2	no	—
174	C₂×M₁₆	—	134	2	no	—
175	C₂×M₁₆	—	134	2	no	—
176	C₂×M₁₆	—	134	1	no	—
177	C₂×M₁₆	Dic₁₃₄	1	4	yes	—
178	C₂²×C₈	—	2	2	no	—
179	C₂²×C₈	—	2	2	no	—
180	C₂²×C₈	—	2	2	no	—
181	C₂²×C₈	—	134	1	no	—
182	C₂²×C₈	C₄×D₆₇	1	2	yes	—
183	C₂²×C₈	D₂₆₈	1	2	yes	—
184	C₂²×C₈	C₂²⋊D₆₇	1	2	yes	—
185	#117	—	134	2	no	—
186	#117	—	134	2	no	—
187	C₂×C₄⋊C₄	—	67	4	no	—
188	C₂×C₄⋊C₄	—	67	2	no	—
189	C₂×C₄⋊C₄	—	134	2	no	—
190	C₂×C₄⋊C₄	—	134	2	no	—
191	C₂×C₄⋊C₄	—	134	2	no	—
192	C₂×C₄⋊C₄	—	134	2	no	—
193	C₂×C₄⋊C₄	—	134	2	no	—
194	C₂×C₂²⋊C₄	—	134	1	no	—
195	C₂×C₄²	—	134	1	no	—
196	#106	—	134	8	no	—
197	C₂⁴	#11761629	1	1	yes	characteristic
198	C₂²×C₄	—	268	2	no	—
199	C₂²×C₄	—	268	2	no	—
200	C₂²×C₄	—	536	4	no	—
201	C₂²×C₄	—	134	2	no	—
202	C₂²×C₄	—	134	4	no	—
203	C₂²×C₄	C₂×C₄×D₆₇	1	1	yes	characteristic
204	C₂²×C₄	D₄×D₆₇	1	1	yes	characteristic
205	C₂²×C₄	D₄×D₆₇	1	1	yes	characteristic
206	C₂²×C₄	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
207	C₂²×C₄	C₂×D₂₆₈	1	1	yes	characteristic
208	C₂²×C₄	C₂×Dic₁₃₄	1	2	yes	—
209	C₂²×C₄	#11761627	1	2	yes	—
210	C₂²×C₄	#11761627	1	2	yes	—
211	C₂²×C₄	#11761628	1	2	yes	—
212	C₂²×C₄	#11761629	1	1	yes	characteristic
213	M₁₆	—	2	4	no	—
214	M₁₆	—	2	4	no	—
215	M₁₆	—	4	2	no	—
216	M₁₆	—	4	2	no	—
217	M₁₆	—	268	2	no	—
218	M₁₆	—	134	4	no	—
219	C₂×C₈	—	2	4	no	—
220	C₂×C₈	—	2	4	no	—
221	C₂×C₈	—	2	2	no	—
222	C₂×C₈	—	2	2	no	—
223	C₂×C₈	—	2	4	no	—
224	C₂×C₈	—	4	2	no	—
225	C₂×C₈	—	4	2	no	—
226	C₂×C₈	—	268	1	no	—
227	C₂×C₈	—	268	1	no	—
228	C₂×C₈	—	134	2	no	—
229	C₂×C₈	(C₂×D₆₇)⋊C₄	1	4	yes	—
230	C₄⋊C₄	—	268	2	no	—
231	C₄⋊C₄	—	268	2	no	—
232	C₄⋊C₄	—	134	4	no	—
233	C₄⋊C₄	—	134	4	no	—
234	C₄⋊C₄	—	134	4	no	—
235	C₂²⋊C₄	—	268	2	no	—
236	C₄²	—	268	1	no	—
237	C₄²	—	268	1	no	—
238	C₂³	—	2	4	no	—
239	C₂³	#14524005	1	1	yes	characteristic
240	C₂³	#14524018	1	1	yes	characteristic
241	C₂³	#14524032	1	1	yes	characteristic
242	C₂³	#14524034	1	2	yes	—
243	C₂³	#14524038	1	2	yes	—
244	C₂×C₄	—	268	4	no	—
245	C₂×C₄	—	268	2	no	—
246	C₂×C₄	—	268	4	no	—
247	C₂×C₄	—	536	4	no	—
248	C₂×C₄	—	536	4	no	—
249	C₂×C₄	—	536	4	no	—
250	C₂×C₄	C₂×(C₂×D₆₇)⋊C₄	1	1	yes	characteristic
251	C₂×C₄	#14523999	1	2	yes	—
252	C₂×C₄	#14524005	1	1	yes	characteristic
253	C₂×C₄	C₄×Dic₁₃₄	1	2	yes	—
254	C₂×C₄	#14524018	1	1	yes	characteristic
255	C₂×C₄	#14524020	1	2	yes	—
256	C₂×C₄	#14524020	1	2	yes	—
257	C₂×C₄	#14524022	1	2	yes	—
258	C₂×C₄	#14524024	1	2	yes	—
259	C₂×C₄	#14524025	1	2	yes	—
260	C₂×C₄	#14524025	1	2	yes	—
261	C₂×C₄	#14524027	1	1	yes	characteristic
262	C₂×C₄	#14524027	1	1	yes	characteristic
263	C₂×C₄	#14524031	1	1	yes	characteristic
264	C₂×C₄	#14524034	1	2	yes	—
265	C₂×C₄	#14524038	1	2	yes	—
266	C₂×C₄	#14524039	1	1	yes	characteristic
267	C₂×C₄	#14524043	1	1	yes	characteristic
268	C₈	—	2	4	no	—
269	C₈	—	4	4	no	—
270	C₈	—	268	2	no	—
271	C₂²	—	2	2	no	—
272	C₂²	—	2	2	no	—
273	C₂²	—	2	4	no	—
274	C₂²	—	2	4	no	—
275	C₂²	#16863279	1	2	yes	—
276	C₂²	#16863297	1	1	yes	characteristic
277	C₂²	#16863323	1	1	yes	characteristic
278	C₂²	#16863328	1	1	yes	characteristic
279	C₂²	#16863335	1	2	yes	—
280	C₂²	#16863660	1	4	yes	—
281	C₄	—	536	4	no	—
282	C₄	—	536	2	no	—
283	C₄	#16863257	1	1	yes	characteristic
284	C₄	#16863279	1	2	yes	—
285	C₄	#16863302	1	1	yes	characteristic
286	C₄	#16863323	1	1	yes	characteristic
287	C₄	#16863328	1	1	yes	characteristic
288	C₄	#16863332	1	2	yes	—
289	C₂	—	2	2	no	—
290	C₂	—	2	2	no	—
291	C₂	#22558478	1	1	yes	characteristic
292	C₂	#22561110	1	2	yes	—
293	C₂	#22561656	1	2	yes	—
294	C₂	#22561721	1	2	yes	—

	subgroup	G/H
center	C₂³	#14524018
commutator	C₂²×C₄×C₆₇	C₂²×C₄
Frattini	C₂³×C₄	C₂²×D₆₇
Fitting	C₂×#341×C₆₇	C₂
socle	C₂³×C₆₇	#121
prime radical	C₂⁴×C₆₇	#41
cosocle	C₂³×C₄×C₆₇	C₂³
hypercenter	C₂×#341	D₆₇
hypocenter	C₆₇	#11366
LCS	G ▷ C₂²×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ C₂×#341

Complements

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

N	H ≅ G/N	type
11. C₂×#14523975	C₄	semidirect
153. C₆₇	#11366	semidirect

Maximal subgroups

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

subgroup	index
C₂²×#14523975	2
C₂×#16863164	2
#22559929	2
#22559905	2
C₂×#341×C₆₇	2
#11366	67