Subgroups — #30864499

Order 17152.

822 conjugacy classes of subgroups.

The subgroup lattice has 260 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16863527	C₂	1	2	yes	maximal
2	#22561146	C₂	1	2	yes	maximal
3	#22560928	C₂	1	4	yes	maximal
4	#22560361	C₂	1	1	yes	maximal, characteristic
5	#22560078	C₂	1	4	yes	maximal
6	C₄²×C₆₇⋊C₈	C₂	1	1	yes	maximal, characteristic
7	C₄×C₄⋊Q₈×C₆₇	C₂	1	1	yes	maximal, characteristic
8	#16863527	—	2	8	no	—
9	C₂×C₄⋊(C₆₇⋊C₈)	C₂²	1	4	yes	—
10	C₂×C₄⋊(C₆₇⋊C₈)	C₂²	1	2	yes	—
11	C₂×C₄⋊(C₆₇⋊C₈)	C₂²	1	4	yes	—
12	C₂×#14523954	C₂²	1	4	yes	—
13	C₂×C₄×C₆₇⋊C₈	C₂²	1	1	yes	characteristic
14	C₂×C₄×C₆₇⋊C₈	C₂²	1	2	yes	—
15	C₂×C₄×C₆₇⋊C₈	C₂²	1	1	yes	characteristic
16	C₂×C₄×C₆₇⋊C₈	C₂²	1	2	yes	—
17	#16863151	C₂²	1	8	yes	—
18	C₂×C₄⋊Q₈×C₆₇	C₄	1	1	yes	characteristic
19	C₂×C₄×Q₈×C₆₇	C₂²	1	2	yes	—
20	#391×C₆₇	C₄	1	2	yes	—
21	#389×C₆₇	C₄	1	4	yes	—
22	#384×C₆₇	C₄	1	1	yes	characteristic
23	C₄×C₄⋊C₄×C₆₇	C₂²	1	4	yes	—
24	C₄³×C₆₇	C₂²	1	1	yes	characteristic
25	C₂²×C₆₇⋊C₈	C₂³	1	4	yes	—
26	C₂²×C₆₇⋊C₈	C₂³	1	1	yes	characteristic
27	C₂²×C₆₇⋊C₈	C₂³	1	2	yes	—
28	C₂²×C₆₇⋊C₈	C₂³	1	1	yes	characteristic
29	C₄⋊(C₆₇⋊C₈)	—	2	8	no	—
30	C₄⋊(C₆₇⋊C₈)	—	2	4	no	—
31	C₄⋊(C₆₇⋊C₈)	—	2	8	no	—
32	#14523954	—	2	8	no	—
33	C₄×C₆₇⋊C₈	—	2	2	no	—
34	C₄×C₆₇⋊C₈	—	2	4	no	—
35	C₄×C₆₇⋊C₈	D₄	1	4	yes	—
36	C₄×C₆₇⋊C₈	Q₈	1	8	yes	complemented
37	C₂²×Q₈×C₆₇	C₂×C₄	1	2	yes	—
38	C₄⋊Q₈×C₆₇	—	2	4	no	—
39	C₄×Q₈×C₆₇	—	2	4	no	—
40	C₄×Q₈×C₆₇	—	2	4	no	—
41	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	2	yes	—
42	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	4	yes	—
43	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	4	yes	—
44	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	4	yes	—
45	C₂×C₄²×C₆₇	C₂×C₄	1	2	yes	—
46	C₂×C₄²×C₆₇	C₂×C₄	1	1	yes	characteristic
47	C₂×C₄²×C₆₇	C₂×C₄	1	1	yes	characteristic
48	C₂×C₄²×C₆₇	C₂³	1	4	yes	—
49	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
50	C₂×C₄²×C₆₇	C₂³	1	2	yes	—
51	#95×C₆₇	C₂×C₄	1	8	yes	—
52	C₂×C₆₇⋊C₈	—	2	2	no	—
53	C₂×C₆₇⋊C₈	—	2	2	no	—
54	C₂×C₆₇⋊C₈	—	2	2	no	—
55	C₂×C₆₇⋊C₈	—	2	4	no	—
56	C₂×C₆₇⋊C₈	—	2	8	no	—
57	C₂×C₆₇⋊C₈	C₂×D₄	1	2	yes	—
58	C₂×C₆₇⋊C₈	C₂×D₄	1	2	yes	—
59	C₂×C₆₇⋊C₈	C₂×Q₈	1	4	yes	—
60	C₂×C₆₇⋊C₈	C₂×Q₈	1	4	yes	—
61	C₂×Q₈×C₆₇	—	2	4	no	—
62	C₂×Q₈×C₆₇	—	2	4	no	—
63	C₂×Q₈×C₆₇	—	2	4	no	—
64	C₂²×C₄×C₆₇	C₂²×C₄	1	4	yes	—
65	C₂²×C₄×C₆₇	C₂²×C₄	1	4	yes	—
66	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
67	C₂²×C₄×C₆₇	C₂²×C₄	1	2	yes	—
68	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
69	C₂²×C₄×C₆₇	C₂²×C₄	1	2	yes	—
70	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
71	C₄⋊C₄×C₆₇	—	2	4	no	—
72	C₄⋊C₄×C₆₇	—	2	8	no	—
73	C₄⋊C₄×C₆₇	—	2	8	no	—
74	C₄⋊C₄×C₆₇	—	2	8	no	—
75	C₄²×C₆₇	—	2	2	no	—
76	C₄²×C₆₇	—	2	4	no	—
77	C₄²×C₆₇	—	2	4	no	—
78	C₄²×C₆₇	—	2	4	no	—
79	C₄²×C₆₇	M₁₆	1	4	yes	—
80	C₄²×C₆₇	C₂×D₄	1	2	yes	—
81	C₄²×C₆₇	C₂×Q₈	1	4	yes	—
82	C₄²×C₆₇	#41	1	2	yes	—
83	C₄²×C₆₇	#41	1	4	yes	—
84	C₂³×C₆₇	C₂³×C₄	1	1	yes	characteristic
85	Q₈×C₆₇	—	4	8	no	—
86	C₂×C₄×C₆₇	—	2	2	no	—
87	C₂×C₄×C₆₇	—	2	2	no	—
88	C₂×C₄×C₆₇	—	2	2	no	—
89	C₂×C₄×C₆₇	—	2	4	no	—
90	C₂×C₄×C₆₇	—	2	4	no	—
91	C₂×C₄×C₆₇	—	2	4	no	—
92	C₂×C₄×C₆₇	—	2	8	no	—
93	C₂×C₄×C₆₇	—	2	4	no	—
94	C₂×C₄×C₆₇	C₄×D₄	1	2	yes	—
95	C₂×C₄×C₆₇	C₄×D₄	1	2	yes	—
96	C₂×C₄×C₆₇	C₄×Q₈	1	4	yes	—
97	C₂×C₄×C₆₇	C₄×Q₈	1	4	yes	—
98	C₂×C₄×C₆₇	C₂×M₁₆	1	2	yes	—
99	C₂×C₄×C₆₇	C₂×M₁₆	1	4	yes	—
100	C₂×C₄×C₆₇	#131	1	2	yes	—
101	C₂×C₄×C₆₇	#131	1	4	yes	—
102	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
103	C₂×C₄×C₆₇	C₂²×Q₈	1	2	yes	—
104	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
105	C₂×C₄×C₆₇	C₂×#41	1	2	yes	—
106	C₆₇⋊C₈	—	2	2	no	—
107	C₆₇⋊C₈	—	2	4	no	—
108	C₆₇⋊C₈	—	4	4	no	—
109	C₆₇⋊C₈	C₄⋊Q₈	1	4	yes	complemented
110	C₂²×C₆₇	C₂×C₄×D₄	1	1	yes	characteristic
111	C₂²×C₆₇	C₂×C₄×Q₈	1	2	yes	—
112	C₂²×C₆₇	C₂²×M₁₆	1	1	yes	characteristic
113	C₂²×C₆₇	C₂×#131	1	1	yes	characteristic
114	C₂²×C₆₇	C₂×#131	1	2	yes	—
115	C₄×C₆₇	—	2	2	no	—
116	C₄×C₆₇	—	2	4	no	—
117	C₄×C₆₇	—	4	4	no	—
118	C₄×C₆₇	—	4	4	no	—
119	C₄×C₆₇	#436	1	4	yes	—
120	C₄×C₆₇	#446	1	8	yes	complemented
121	C₄×C₆₇	C₂×C₄⋊Q₈	1	1	yes	characteristic
122	C₄×C₆₇	#532	1	1	yes	characteristic
123	C₄×C₆₇	#533	1	2	yes	—
124	#21712	—	67	1	no	maximal, Sylow
125	C₂×C₆₇	C₄×C₄⋊Q₈	1	1	yes	characteristic
126	C₂×C₆₇	C₂×#436	1	1	yes	characteristic
127	C₂×C₆₇	#2931	1	1	yes	characteristic
128	C₂×C₆₇	C₂×#446	1	2	yes	—
129	C₂×C₆₇	#2960	1	2	yes	—
130	C₂×#446	—	67	2	no	—
131	C₄×C₄⋊Q₈	D₆₇	1	1	yes	characteristic
132	#1984	—	67	2	no	—
133	#1939	—	67	4	no	—
134	#1842	—	67	1	no	—
135	#1770	—	67	4	no	—
136	C₄²×C₈	—	67	1	no	—
137	C₆₇	#21712	1	1	yes	characteristic, complemented, Sylow
138	C₂×C₄⋊Q₈	Dic₆₇	1	1	yes	characteristic
139	C₂×C₄×Q₈	C₂×D₆₇	1	2	yes	—
140	#446	—	134	8	no	—
141	C₂×C₄⋊C₈	—	67	2	no	—
142	C₂×C₄⋊C₈	—	67	4	no	—
143	C₂×C₄⋊C₈	—	67	4	no	—
144	C₂×#97	—	67	4	no	—
145	C₂×C₄×C₈	—	67	1	no	—
146	C₂×C₄×C₈	—	67	2	no	—
147	C₂×C₄×C₈	—	67	2	no	—
148	C₂×C₄×C₈	—	67	1	no	—
149	#391	Dic₆₇	1	2	yes	—
150	#389	Dic₆₇	1	4	yes	—
151	#384	Dic₆₇	1	1	yes	characteristic
152	C₄×C₄⋊C₄	C₂×D₆₇	1	4	yes	—
153	C₄³	C₂×D₆₇	1	1	yes	characteristic
154	#336	—	67	8	no	—
155	C₂²×Q₈	C₂×Dic₆₇	1	2	yes	—
156	C₂²×C₈	—	67	1	no	—
157	C₂²×C₈	—	67	2	no	—
158	C₂²×C₈	—	67	1	no	—
159	C₂²×C₈	—	67	4	no	—
160	C₄⋊Q₈	—	2	4	no	—
161	C₄×Q₈	—	2	4	no	—
162	C₄×Q₈	—	2	4	no	—
163	C₂×C₄⋊C₄	C₂×Dic₆₇	1	2	yes	—
164	C₂×C₄⋊C₄	C₂×Dic₆₇	1	4	yes	—
165	C₂×C₄⋊C₄	C₂×Dic₆₇	1	4	yes	—
166	C₂×C₄⋊C₄	C₂×Dic₆₇	1	4	yes	—
167	C₂×C₄²	C₂×Dic₆₇	1	2	yes	—
168	C₂×C₄²	C₂×Dic₆₇	1	1	yes	characteristic
169	C₂×C₄²	C₂×Dic₆₇	1	1	yes	characteristic
170	C₂×C₄²	C₂²×D₆₇	1	4	yes	—
171	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
172	C₂×C₄²	C₂²×D₆₇	1	2	yes	—
173	C₄⋊C₈	—	134	8	no	—
174	C₄⋊C₈	—	134	4	no	—
175	C₄⋊C₈	—	134	8	no	—
176	#97	—	134	8	no	—
177	C₄×C₈	—	67	4	no	—
178	C₄×C₈	—	67	8	no	—
179	C₄×C₈	—	134	2	no	—
180	C₄×C₈	—	134	4	no	—
181	#95	C₂×Dic₆₇	1	8	yes	—
182	C₂×Q₈	—	2	4	no	—
183	C₂×Q₈	—	2	4	no	—
184	C₂×Q₈	—	2	4	no	—
185	C₂²×C₄	C₂²×Dic₆₇	1	4	yes	—
186	C₂²×C₄	C₂²×Dic₆₇	1	4	yes	—
187	C₂²×C₄	C₂²×Dic₆₇	1	1	yes	characteristic
188	C₂²×C₄	C₂²×Dic₆₇	1	2	yes	—
189	C₂²×C₄	C₂²×Dic₆₇	1	1	yes	characteristic
190	C₂²×C₄	C₂²×Dic₆₇	1	2	yes	—
191	C₂²×C₄	C₂³×D₆₇	1	1	yes	characteristic
192	C₂×C₈	—	67	2	no	—
193	C₂×C₈	—	67	4	no	—
194	C₂×C₈	—	67	2	no	—
195	C₂×C₈	—	67	4	no	—
196	C₂×C₈	—	134	4	no	—
197	C₂×C₈	—	134	8	no	—
198	C₂×C₈	—	134	2	no	—
199	C₂×C₈	—	134	2	no	—
200	C₂×C₈	—	134	2	no	—
201	C₄⋊C₄	—	2	4	no	—
202	C₄⋊C₄	—	2	8	no	—
203	C₄⋊C₄	—	2	8	no	—
204	C₄⋊C₄	—	2	8	no	—
205	C₄²	—	2	4	no	—
206	C₄²	—	2	2	no	—
207	C₄²	—	2	4	no	—
208	C₄²	—	2	4	no	—
209	C₄²	#11761612	1	4	yes	—
210	C₄²	D₄×D₆₇	1	2	yes	—
211	C₄²	Q₈×D₆₇	1	4	yes	—
212	C₄²	#11761627	1	2	yes	—
213	C₄²	#11761629	1	4	yes	—
214	C₂³	C₂³×Dic₆₇	1	1	yes	characteristic
215	Q₈	—	4	8	no	—
216	C₂×C₄	—	2	2	no	—
217	C₂×C₄	—	2	2	no	—
218	C₂×C₄	—	2	2	no	—
219	C₂×C₄	—	2	4	no	—
220	C₂×C₄	—	2	4	no	—
221	C₂×C₄	—	2	4	no	—
222	C₂×C₄	—	2	8	no	—
223	C₂×C₄	—	2	4	no	—
224	C₂×C₄	D₄×Dic₆₇	1	2	yes	—
225	C₂×C₄	D₄×Dic₆₇	1	2	yes	—
226	C₂×C₄	Q₈×Dic₆₇	1	4	yes	—
227	C₂×C₄	Q₈×Dic₆₇	1	4	yes	—
228	C₂×C₄	C₂×#11761612	1	4	yes	—
229	C₂×C₄	C₂×#11761612	1	2	yes	—
230	C₂×C₄	#14524050	1	4	yes	—
231	C₂×C₄	#14524050	1	2	yes	—
232	C₂×C₄	C₂×D₄×D₆₇	1	1	yes	characteristic
233	C₂×C₄	C₂×Q₈×D₆₇	1	2	yes	—
234	C₂×C₄	C₂×#11761627	1	1	yes	characteristic
235	C₂×C₄	C₂×#11761629	1	2	yes	—
236	C₈	—	268	4	no	—
237	C₈	—	67	4	no	—
238	C₈	—	134	2	no	—
239	C₈	—	134	4	no	—
240	C₂²	C₂×D₄×Dic₆₇	1	1	yes	characteristic
241	C₂²	C₂×Q₈×Dic₆₇	1	2	yes	—
242	C₂²	C₂²×#11761612	1	1	yes	characteristic
243	C₂²	C₂×#14524050	1	2	yes	—
244	C₂²	C₂×#14524050	1	1	yes	characteristic
245	C₄	—	2	4	no	—
246	C₄	—	2	2	no	—
247	C₄	—	4	4	no	—
248	C₄	—	4	4	no	—
249	C₄	#16863474	1	4	yes	—
250	C₄	#16863527	1	8	yes	complemented
251	C₄	C₄⋊Q₈×D₆₇	1	1	yes	characteristic
252	C₄	#16863947	1	1	yes	characteristic
253	C₄	#16863951	1	2	yes	—
254	C₂	C₄⋊Q₈×Dic₆₇	1	1	yes	characteristic
255	C₂	C₂×#16863474	1	1	yes	characteristic
256	C₂	#22568921	1	1	yes	characteristic
257	C₂	C₂×#16863527	1	2	yes	—
258	C₂	#22569144	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂³×D₆₇
commutator, socle, prime radical	C₂³×C₆₇	C₂³×C₄
Fitting	C₄×C₄⋊Q₈×C₆₇	C₂
cosocle	C₂²×C₄×C₆₇	C₂⁴
hypercenter	C₄×C₄⋊Q₈	D₆₇
hypocenter	C₆₇	#21712
LCS	G ▷ C₂³×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂²×C₄ ◁ C₄×C₄⋊Q₈

Complements

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

N	H ≅ G/N	type
36. C₄×C₆₇⋊C₈	Q₈	semidirect
109. C₆₇⋊C₈	C₄⋊Q₈	semidirect
120. C₄×C₆₇	#446	semidirect
137. C₆₇	#21712	semidirect
250. C₄	#16863527	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16863527	2
#22561146	2
#22560928	2
#22560361	2
#22560078	2
C₄²×C₆₇⋊C₈	2
C₄×C₄⋊Q₈×C₆₇	2
#21712	67