Subgroups — #30764266

Order 17152.

574 conjugacy classes of subgroups.

The subgroup lattice has 284 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16863167	C₂	1	2	yes	maximal
2	C₂×#16863160	C₂	1	2	yes	maximal
3	C₂×C₆₇⋊#338	C₂	1	1	yes	maximal, characteristic
4	#22559907	C₂	1	1	yes	maximal, characteristic
5	C₂×#430×C₆₇	C₂	1	1	yes	maximal, characteristic
6	C₂²×#11761612	C₂²	1	2	yes	—
7	C₂³×C₆₇⋊C₈	C₂²	1	1	yes	characteristic
8	C₂×#14524002	C₂²	1	1	yes	characteristic
9	#16863167	—	2	4	no	—
10	#16863160	—	2	4	no	—
11	C₆₇⋊#338	C₄	1	4	yes	complemented
12	#16863150	—	2	2	no	—
13	C₂²×M₁₆×C₆₇	C₂²	1	2	yes	—
14	C₂²×M₁₆×C₆₇	C₂²	1	1	yes	characteristic
15	#430×C₆₇	—	2	4	no	—
16	C₂×#108×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	1	no	—
24	C₂²×C₆₇⋊C₈	—	2	1	no	—
25	C₂²×C₆₇⋊C₈	—	2	2	no	—
26	C₂²×C₆₇⋊C₈	—	2	2	no	—
27	C₂²×C₆₇⋊C₈	—	2	2	no	—
28	C₂²×C₆₇⋊C₈	D₄	1	2	yes	—
29	#14524002	—	2	2	no	—
30	#14524002	—	2	2	no	—
31	C₂×C₄⋊Dic₆₇	—	2	1	no	—
32	C₂×C₂²⋊Dic₆₇	Q₈	1	2	yes	—
33	C₂×C₄×Dic₆₇	—	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	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	C₂×C₄	1	2	yes	—
43	C₂×M₁₆×C₆₇	D₄	1	4	yes	—
44	C₂×M₁₆×C₆₇	Q₈	1	2	yes	—
45	C₂²×C₈×C₆₇	—	2	2	no	—
46	C₂²×C₈×C₆₇	D₄	1	2	yes	—
47	#108×C₆₇	—	2	8	no	—
48	C₂²×Dic₆₇	—	2	2	no	—
49	#11761612	—	2	4	no	—
50	#11761612	—	2	4	no	—
51	#11761612	—	4	2	no	—
52	#11761612	—	4	2	no	—
53	C₂×C₆₇⋊C₈	—	2	4	no	—
54	C₂×C₆₇⋊C₈	—	2	2	no	—
55	C₂×C₆₇⋊C₈	—	2	4	no	—
56	C₂×C₆₇⋊C₈	—	4	2	no	—
57	C₂×C₆₇⋊C₈	—	4	1	no	—
58	C₂×C₆₇⋊C₈	—	4	1	no	—
59	C₂×C₆₇⋊C₈	—	4	2	no	—
60	C₂×C₆₇⋊C₈	—	4	2	no	—
61	C₄⋊Dic₆₇	—	4	2	no	—
62	C₂²⋊Dic₆₇	—	2	4	no	—
63	C₄×Dic₆₇	—	4	1	no	—
64	C₄×Dic₆₇	—	4	1	no	—
65	C₂⁴×C₆₇	C₂×Q₈	1	1	yes	characteristic
66	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
67	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
68	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
69	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
70	C₂²×C₄×C₆₇	C₂×Q₈	1	1	yes	characteristic
71	C₂²×C₄×C₆₇	#41	1	2	yes	—
72	C₂²×C₄×C₆₇	#41	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	M₁₆×C₆₇	—	2	4	no	—
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	M₁₆×C₆₇	—	4	2	no	—
82	C₂×C₈×C₆₇	—	2	2	no	—
83	C₂×C₈×C₆₇	—	2	2	no	—
84	C₂×C₈×C₆₇	—	2	4	no	—
85	C₂×C₈×C₆₇	—	4	2	no	—
86	C₂×C₈×C₆₇	—	4	2	no	—
87	C₂×C₈×C₆₇	C₄⋊C₄	1	4	yes	—
88	C₂³×C₆₇	—	4	2	no	—
89	C₂³×C₆₇	C₄×Q₈	1	1	yes	characteristic
90	C₂³×C₆₇	C₂²⋊Q₈	1	1	yes	characteristic
91	C₂³×C₆₇	C₂²⋊Q₈	1	2	yes	—
92	C₂³×C₆₇	#125	1	2	yes	—
93	C₂³×C₆₇	C₄⋊Q₈	1	1	yes	characteristic
94	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
95	C₂×C₄×C₆₇	#117	1	2	yes	—
96	C₂×C₄×C₆₇	C₄×D₄	1	2	yes	—
97	C₂×C₄×C₆₇	C₄×D₄	1	1	yes	characteristic
98	C₂×C₄×C₆₇	#120	1	1	yes	characteristic
99	C₂×C₄×C₆₇	#121	1	1	yes	characteristic
100	C₂×C₄×C₆₇	#121	1	2	yes	—
101	C₂×C₄×C₆₇	C₂²⋊Q₈	1	2	yes	—
102	C₂×C₄×C₆₇	#123	1	2	yes	—
103	C₂×C₄×C₆₇	#123	1	1	yes	characteristic
104	C₂×C₄×C₆₇	#123	1	2	yes	—
105	C₂×C₄×C₆₇	#124	1	1	yes	characteristic
106	C₂×C₄×C₆₇	#124	1	1	yes	characteristic
107	C₂×C₄×C₆₇	#124	1	2	yes	—
108	C₂×C₄×C₆₇	#125	1	2	yes	—
109	C₂×C₄×C₆₇	#125	1	1	yes	characteristic
110	C₂×C₄×C₆₇	#126	1	2	yes	—
111	C₂×C₄×C₆₇	#126	1	2	yes	—
112	C₂×Dic₆₇	—	4	4	no	—
113	C₂×Dic₆₇	—	4	2	no	—
114	C₈×C₆₇	—	2	4	no	—
115	C₈×C₆₇	—	4	4	no	—
116	C₆₇⋊C₈	—	4	4	no	—
117	C₆₇⋊C₈	—	4	2	no	—
118	C₂²×C₆₇	—	2	2	no	—
119	C₂²×C₆₇	—	2	4	no	—
120	C₂²×C₆₇	—	4	1	no	—
121	C₂²×C₆₇	—	4	2	no	—
122	C₂²×C₆₇	—	4	1	no	—
123	C₂²×C₆₇	#387	1	2	yes	—
124	C₂²×C₆₇	#389	1	1	yes	characteristic
125	C₂²×C₆₇	#393	1	1	yes	characteristic
126	C₂²×C₆₇	#398	1	2	yes	—
127	C₂²×C₆₇	#400	1	1	yes	characteristic
128	C₄×C₆₇	#382	1	1	yes	characteristic
129	C₄×C₆₇	#385	1	1	yes	characteristic
130	C₄×C₆₇	#388	1	2	yes	—
131	C₄×C₆₇	#394	1	1	yes	characteristic
132	C₄×C₆₇	#397	1	1	yes	characteristic
133	C₄×C₆₇	#400	1	2	yes	—
134	Dic₆₇	—	8	2	no	—
135	#11295	—	67	1	no	maximal, Sylow
136	C₂×C₆₇	—	2	1	no	—
137	C₂×C₆₇	—	2	2	no	—
138	C₂×C₆₇	—	2	1	no	—
139	C₂×C₆₇	—	2	2	no	—
140	C₂×C₆₇	#1442	1	1	yes	characteristic
141	C₂×C₆₇	#2103	1	2	yes	—
142	C₂×#430	D₆₇	1	1	yes	characteristic
143	C₂×#344	—	67	2	no	—
144	C₂×#341	—	67	2	no	—
145	C₂×#338	—	67	1	no	—
146	#1729	—	67	1	no	—
147	C₆₇	#11295	1	1	yes	characteristic, complemented, Sylow
148	C₂²×M₁₆	—	67	2	no	—
149	C₂²×M₁₆	C₂×D₆₇	1	2	yes	—
150	C₂²×M₁₆	C₂×D₆₇	1	1	yes	characteristic
151	C₂³×C₈	—	67	1	no	—
152	C₂×#117	—	67	1	no	—
153	#430	—	2	4	no	—
154	C₂×#108	—	2	2	no	—
155	#344	—	134	4	no	—
156	#341	—	134	4	no	—
157	#338	—	67	4	no	—
158	#336	—	134	2	no	—
159	C₂³×C₄	C₂²×D₆₇	1	1	yes	characteristic
160	C₂×M₁₆	—	2	2	no	—
161	C₂×M₁₆	—	2	2	no	—
162	C₂×M₁₆	—	2	2	no	—
163	C₂×M₁₆	—	2	2	no	—
164	C₂×M₁₆	—	2	2	no	—
165	C₂×M₁₆	—	2	2	no	—
166	C₂×M₁₆	—	2	2	no	—
167	C₂×M₁₆	—	67	4	no	—
168	C₂×M₁₆	—	134	2	no	—
169	C₂×M₁₆	—	134	2	no	—
170	C₂×M₁₆	—	134	2	no	—
171	C₂×M₁₆	—	134	2	no	—
172	C₂×M₁₆	—	134	2	no	—
173	C₂×M₁₆	Dic₁₃₄	1	2	yes	—
174	C₂×M₁₆	C₄×D₆₇	1	2	yes	—
175	C₂×M₁₆	C₂²⋊D₆₇	1	4	yes	—
176	C₂²×C₈	—	2	2	no	—
177	C₂²×C₈	—	67	2	no	—
178	C₂²×C₈	—	134	1	no	—
179	C₂²×C₈	—	134	1	no	—
180	C₂²×C₈	—	134	2	no	—
181	C₂²×C₈	—	134	2	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	C₂×C₄⋊C₄	—	134	1	no	—
187	C₂×C₂²⋊C₄	—	67	2	no	—
188	C₂×C₄²	—	134	1	no	—
189	#108	—	2	8	no	—
190	C₂⁴	Q₈×D₆₇	1	1	yes	characteristic
191	C₂²×C₄	—	134	2	no	—
192	C₂²×C₄	C₂×C₄×D₆₇	1	1	yes	characteristic
193	C₂²×C₄	D₄×D₆₇	1	1	yes	characteristic
194	C₂²×C₄	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
195	C₂²×C₄	C₂×C₂²⋊D₆₇	1	2	yes	—
196	C₂²×C₄	C₂×Dic₁₃₄	1	1	yes	characteristic
197	C₂²×C₄	#11761627	1	2	yes	—
198	C₂²×C₄	#11761627	1	1	yes	characteristic
199	C₂²×C₄	#11761628	1	2	yes	—
200	C₂²×C₄	#11761629	1	2	yes	—
201	C₂²×C₄	#11761629	1	1	yes	characteristic
202	M₁₆	—	2	4	no	—
203	M₁₆	—	2	4	no	—
204	M₁₆	—	2	4	no	—
205	M₁₆	—	2	4	no	—
206	M₁₆	—	4	2	no	—
207	M₁₆	—	4	2	no	—
208	M₁₆	—	268	2	no	—
209	M₁₆	—	268	2	no	—
210	M₁₆	—	134	4	no	—
211	M₁₆	—	134	4	no	—
212	C₂×C₈	—	2	2	no	—
213	C₂×C₈	—	2	2	no	—
214	C₂×C₈	—	2	4	no	—
215	C₂×C₈	—	4	2	no	—
216	C₂×C₈	—	4	2	no	—
217	C₂×C₈	—	268	2	no	—
218	C₂×C₈	—	268	1	no	—
219	C₂×C₈	—	268	1	no	—
220	C₂×C₈	—	268	2	no	—
221	C₂×C₈	—	268	2	no	—
222	C₂×C₈	—	134	4	no	—
223	C₂×C₈	—	134	2	no	—
224	C₂×C₈	—	134	4	no	—
225	C₂×C₈	Dic₆₇⋊C₄	1	4	yes	—
226	C₄⋊C₄	—	268	2	no	—
227	C₂²⋊C₄	—	134	4	no	—
228	C₄²	—	268	1	no	—
229	C₄²	—	268	1	no	—
230	C₂³	—	4	2	no	—
231	C₂³	#14524008	1	1	yes	characteristic
232	C₂³	#14524019	1	1	yes	characteristic
233	C₂³	#14524019	1	2	yes	—
234	C₂³	#14524033	1	2	yes	—
235	C₂³	#14524042	1	1	yes	characteristic
236	C₂×C₄	—	268	4	no	—
237	C₂×C₄	—	268	2	no	—
238	C₂×C₄	C₂×Dic₆₇⋊C₄	1	1	yes	characteristic
239	C₂×C₄	#14524001	1	2	yes	—
240	C₂×C₄	C₄×C₂²⋊D₆₇	1	2	yes	—
241	C₂×C₄	#14524005	1	1	yes	characteristic
242	C₂×C₄	#14524012	1	1	yes	characteristic
243	C₂×C₄	#14524014	1	1	yes	characteristic
244	C₂×C₄	#14524015	1	2	yes	—
245	C₂×C₄	#14524022	1	2	yes	—
246	C₂×C₄	#14524024	1	2	yes	—
247	C₂×C₄	#14524026	1	1	yes	characteristic
248	C₂×C₄	#14524028	1	2	yes	—
249	C₂×C₄	#14524030	1	1	yes	characteristic
250	C₂×C₄	#14524032	1	1	yes	characteristic
251	C₂×C₄	#14524032	1	2	yes	—
252	C₂×C₄	#14524034	1	2	yes	—
253	C₂×C₄	#14524034	1	1	yes	characteristic
254	C₂×C₄	#14524038	1	2	yes	—
255	C₂×C₄	#14524038	1	2	yes	—
256	C₈	—	2	4	no	—
257	C₈	—	4	4	no	—
258	C₈	—	268	4	no	—
259	C₈	—	268	2	no	—
260	C₂²	—	2	2	no	—
261	C₂²	—	2	4	no	—
262	C₂²	—	4	1	no	—
263	C₂²	—	4	2	no	—
264	C₂²	—	4	1	no	—
265	C₂²	#16863277	1	2	yes	—
266	C₂²	#16863292	1	1	yes	characteristic
267	C₂²	#16863307	1	1	yes	characteristic
268	C₂²	#16863329	1	2	yes	—
269	C₂²	#16863334	1	1	yes	characteristic
270	C₄	—	536	2	no	—
271	C₄	#16863261	1	1	yes	characteristic
272	C₄	#16863269	1	1	yes	characteristic
273	C₄	#16863286	1	2	yes	—
274	C₄	#16863310	1	1	yes	characteristic
275	C₄	#16863322	1	1	yes	characteristic
276	C₄	#16863335	1	2	yes	—
277	C₂	—	2	1	no	—
278	C₂	—	2	2	no	—
279	C₂	—	2	1	no	—
280	C₂	—	2	2	no	—
281	C₂	#22558468	1	1	yes	characteristic
282	C₂	#22561788	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	#14524030
commutator	C₂²×C₄×C₆₇	C₂²×C₄
Frattini	C₂³×C₄	C₂²×D₆₇
Fitting	C₂×#430×C₆₇	C₂
socle	C₂²×C₆₇	#400
prime radical	C₂⁴×C₆₇	C₂×Q₈
cosocle	C₂³×C₄×C₆₇	C₂³
hypercenter	C₂×#430	D₆₇
hypocenter	C₆₇	#11295
LCS	G ▷ C₂²×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ C₂×#430

Complements

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

N	H ≅ G/N	type
11. C₆₇⋊#338	C₄	semidirect
147. C₆₇	#11295	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16863167	2
C₂×#16863160	2
C₂×C₆₇⋊#338	2
#22559907	2
C₂×#430×C₆₇	2
#11295	67