Subgroups — #31009340

Order 17152.

498 conjugacy classes of subgroups.

The subgroup lattice has 256 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22562118	C₂	1	2	yes	maximal, complemented
2	#22562118	C₂	1	4	yes	maximal, complemented
3	Q₈×C₆₇⋊C₁₆	C₂	1	2	yes	maximal, complemented
4	#22561968	C₂	1	2	yes	maximal
5	C₂×C₄⋊(C₆₇⋊C₁₆)	C₂	1	1	yes	maximal, characteristic
6	#22561806	C₂	1	2	yes	maximal
7	C₄×#14523983	C₂	1	1	yes	maximal, characteristic
8	C₆₇×#2960	C₂	1	1	yes	maximal, characteristic
9	C₂×#14523983	C₂²	1	1	yes	characteristic
10	C₂×#14523983	C₂²	1	1	yes	characteristic
11	C₂²×C₆₇⋊C₁₆	C₂²	1	2	yes	—
12	C₄⋊(C₆₇⋊C₁₆)	C₂²	1	1	yes	characteristic
13	C₄⋊(C₆₇⋊C₁₆)	C₂²	1	4	yes	—
14	C₄⋊(C₆₇⋊C₁₆)	C₂²	1	2	yes	—
15	C₄⋊(C₆₇⋊C₁₆)	C₂²	1	1	yes	characteristic
16	C₄⋊(C₆₇⋊C₁₆)	C₂²	1	4	yes	—
17	#16863172	C₂²	1	2	yes	—
18	#16863172	C₂²	1	4	yes	—
19	C₄×C₆₇⋊C₁₆	C₂²	1	2	yes	—
20	C₄×C₆₇⋊C₁₆	C₂²	1	4	yes	—
21	#533×C₆₇	C₄	1	1	yes	characteristic
22	#446×C₆₇	C₄	1	2	yes	—
23	#446×C₆₇	C₄	1	2	yes	—
24	C₈×Q₈×C₆₇	C₂²	1	2	yes	—
25	C₈×Q₈×C₆₇	C₂²	1	2	yes	—
26	#424×C₆₇	C₄	1	2	yes	—
27	#423×C₆₇	C₄	1	1	yes	characteristic
28	#405×C₆₇	C₂²	1	2	yes	—
29	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
30	#14523983	—	2	2	no	—
31	#14523983	Q₈	1	4	yes	complemented
32	C₂×C₆₇⋊C₁₆	—	2	4	no	—
33	C₂×C₆₇⋊C₁₆	C₂³	1	2	yes	—
34	C₂×C₆₇⋊C₁₆	C₂³	1	2	yes	—
35	C₂×C₆₇⋊C₁₆	C₂³	1	2	yes	—
36	C₂×C₆₇⋊C₁₆	C₂³	1	2	yes	—
37	C₂×M₁₆×C₆₇	C₂×C₄	1	2	yes	—
38	C₂²×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
39	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
40	C₄⋊Q₈×C₆₇	C₈	1	1	yes	characteristic
41	C₄⋊Q₈×C₆₇	C₈	1	1	yes	characteristic
42	#125×C₆₇	C₈	1	2	yes	—
43	C₂²⋊Q₈×C₆₇	C₈	1	2	yes	—
44	C₂²⋊Q₈×C₆₇	C₈	1	2	yes	—
45	C₄×Q₈×C₆₇	C₂×C₄	1	2	yes	—
46	C₄×Q₈×C₆₇	C₂×C₄	1	2	yes	—
47	#117×C₆₇	C₂×C₄	1	2	yes	—
48	C₂×C₄²×C₆₇	C₂×C₄	1	1	yes	characteristic
49	C₄⋊C₈×C₆₇	C₂×C₄	1	2	yes	—
50	C₄⋊C₈×C₆₇	C₂×C₄	1	2	yes	—
51	C₄⋊C₈×C₆₇	C₂×C₄	1	4	yes	—
52	C₄⋊C₈×C₆₇	C₂×C₄	1	4	yes	—
53	#97×C₆₇	C₂×C₄	1	4	yes	—
54	C₄×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
55	C₄×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
56	C₄×C₈×C₆₇	C₂³	1	1	yes	characteristic
57	C₄×C₈×C₆₇	C₂³	1	4	yes	—
58	C₄×C₈×C₆₇	C₂³	1	1	yes	characteristic
59	C₆₇⋊C₁₆	—	2	2	no	—
60	C₆₇⋊C₁₆	—	2	2	no	—
61	C₆₇⋊C₁₆	—	2	2	no	—
62	C₆₇⋊C₁₆	C₂×Q₈	1	4	yes	—
63	C₂×Q₈×C₆₇	C₂×C₈	1	2	yes	—
64	C₂×Q₈×C₆₇	C₂×C₈	1	2	yes	—
65	C₂²×C₄×C₆₇	C₂×C₈	1	1	yes	characteristic
66	C₂²×C₄×C₆₇	C₂×C₈	1	1	yes	characteristic
67	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
68	M₁₆×C₆₇	—	2	4	no	—
69	C₂×C₈×C₆₇	—	2	2	no	—
70	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
71	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
72	C₂×C₈×C₆₇	C₂²×C₄	1	4	yes	—
73	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
74	C₂×C₈×C₆₇	C₂×Q₈	1	2	yes	—
75	C₂×C₈×C₆₇	#41	1	2	yes	—
76	C₂×C₈×C₆₇	C₂⁴	1	1	yes	characteristic
77	C₄⋊C₄×C₆₇	C₂×C₈	1	2	yes	—
78	C₄⋊C₄×C₆₇	C₂×C₈	1	2	yes	—
79	C₄⋊C₄×C₆₇	C₂×C₈	1	4	yes	—
80	C₄⋊C₄×C₆₇	C₂×C₈	1	4	yes	—
81	C₄⋊C₄×C₆₇	C₂×C₈	1	2	yes	—
82	C₄⋊C₄×C₆₇	C₂×C₈	1	2	yes	—
83	C₂²⋊C₄×C₆₇	C₂×C₈	1	4	yes	—
84	C₄²×C₆₇	C₂×C₈	1	1	yes	characteristic
85	C₄²×C₆₇	C₂×C₈	1	1	yes	characteristic
86	C₄²×C₆₇	C₂²×C₄	1	1	yes	characteristic
87	C₄²×C₆₇	C₂²×C₄	1	1	yes	characteristic
88	C₄²×C₆₇	C₂²×C₄	1	4	yes	—
89	C₂³×C₆₇	C₂²×C₈	1	1	yes	characteristic
90	Q₈×C₆₇	—	2	4	no	—
91	Q₈×C₆₇	—	4	2	no	—
92	C₂×C₄×C₆₇	—	2	2	no	—
93	C₂×C₄×C₆₇	—	4	1	no	—
94	C₂×C₄×C₆₇	C₄×Q₈	1	2	yes	—
95	C₂×C₄×C₆₇	C₂²×C₈	1	1	yes	characteristic
96	C₂×C₄×C₆₇	C₂²×C₈	1	1	yes	characteristic
97	C₂×C₄×C₆₇	C₂²×C₈	1	1	yes	characteristic
98	C₂×C₄×C₆₇	C₂²×C₈	1	4	yes	—
99	C₂×C₄×C₆₇	C₂²×C₈	1	4	yes	—
100	C₂×C₄×C₆₇	C₂²×C₈	1	1	yes	characteristic
101	C₂×C₄×C₆₇	C₂²×C₈	1	1	yes	characteristic
102	C₂×C₄×C₆₇	#131	1	2	yes	—
103	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
104	C₈×C₆₇	—	2	1	no	—
105	C₈×C₆₇	—	2	1	no	—
106	C₈×C₆₇	—	2	4	no	—
107	C₈×C₆₇	C₄×Q₈	1	2	yes	—
108	C₈×C₆₇	C₂²×Q₈	1	1	yes	characteristic
109	C₈×C₆₇	C₂×#41	1	1	yes	characteristic
110	C₂²×C₆₇	—	2	1	no	—
111	C₂²×C₆₇	—	2	1	no	—
112	C₂²×C₆₇	C₈×Q₈	1	2	yes	—
113	C₂²×C₆₇	C₂³×C₈	1	1	yes	characteristic
114	C₄×C₆₇	—	2	1	no	—
115	C₄×C₆₇	—	2	4	no	—
116	C₄×C₆₇	—	2	1	no	—
117	C₄×C₆₇	—	2	1	no	—
118	C₄×C₆₇	—	2	4	no	—
119	C₄×C₆₇	C₈×Q₈	1	2	yes	—
120	C₄×C₆₇	#504	1	2	yes	—
121	C₄×C₆₇	C₂×C₄×Q₈	1	1	yes	characteristic
122	C₄×C₆₇	C₂×#131	1	1	yes	characteristic
123	#33476	—	67	1	no	maximal, Sylow
124	C₂×C₆₇	—	2	2	no	—
125	C₂×C₆₇	C₂×C₈×Q₈	1	1	yes	characteristic
126	C₂×C₆₇	C₂×#504	1	1	yes	characteristic
127	C₂×C₆₇	#3407	1	1	yes	characteristic
128	#2960	D₆₇	1	1	yes	characteristic
129	#2183	—	67	2	no	—
130	#2183	—	67	4	no	—
131	Q₈×C₁₆	—	67	2	no	—
132	#2151	—	67	2	no	—
133	C₂×C₄⋊C₁₆	—	67	1	no	—
134	#2108	—	67	2	no	—
135	C₄×M₃₂	—	67	1	no	—
136	C₆₇	#33476	1	1	yes	characteristic, complemented, Sylow
137	#533	Dic₆₇	1	1	yes	characteristic
138	C₂×M₃₂	—	67	1	no	—
139	C₂×M₃₂	—	67	1	no	—
140	C₂²×C₁₆	—	67	2	no	—
141	#446	Dic₆₇	1	2	yes	—
142	#446	Dic₆₇	1	2	yes	—
143	C₈×Q₈	C₂×D₆₇	1	2	yes	—
144	C₈×Q₈	C₂×D₆₇	1	2	yes	—
145	#424	Dic₆₇	1	2	yes	—
146	#423	Dic₆₇	1	1	yes	characteristic
147	#405	C₂×D₆₇	1	2	yes	—
148	C₂×C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
149	C₄⋊C₁₆	—	67	1	no	—
150	C₄⋊C₁₆	—	67	1	no	—
151	C₄⋊C₁₆	—	67	2	no	—
152	C₄⋊C₁₆	—	67	4	no	—
153	C₄⋊C₁₆	—	67	4	no	—
154	#346	—	67	2	no	—
155	#346	—	67	4	no	—
156	C₄×C₁₆	—	67	2	no	—
157	C₄×C₁₆	—	67	4	no	—
158	C₂×M₁₆	C₂×Dic₆₇	1	2	yes	—
159	C₂²×C₈	C₂×Dic₆₇	1	1	yes	characteristic
160	C₂²×C₈	C₂²×D₆₇	1	1	yes	characteristic
161	C₄⋊Q₈	C₆₇⋊C₈	1	1	yes	characteristic
162	C₄⋊Q₈	C₆₇⋊C₈	1	1	yes	characteristic
163	#125	C₆₇⋊C₈	1	2	yes	—
164	C₂²⋊Q₈	C₆₇⋊C₈	1	2	yes	—
165	C₂²⋊Q₈	C₆₇⋊C₈	1	2	yes	—
166	C₄×Q₈	C₂×Dic₆₇	1	2	yes	—
167	C₄×Q₈	C₂×Dic₆₇	1	2	yes	—
168	#117	C₂×Dic₆₇	1	2	yes	—
169	C₂×C₄²	C₂×Dic₆₇	1	1	yes	characteristic
170	M₃₂	—	67	4	no	—
171	M₃₂	—	134	2	no	—
172	C₂×C₁₆	—	67	2	no	—
173	C₂×C₁₆	—	67	2	no	—
174	C₂×C₁₆	—	67	2	no	—
175	C₂×C₁₆	—	67	2	no	—
176	C₂×C₁₆	—	134	4	no	—
177	C₄⋊C₈	C₂×Dic₆₇	1	2	yes	—
178	C₄⋊C₈	C₂×Dic₆₇	1	2	yes	—
179	C₄⋊C₈	C₂×Dic₆₇	1	4	yes	—
180	C₄⋊C₈	C₂×Dic₆₇	1	4	yes	—
181	#97	C₂×Dic₆₇	1	4	yes	—
182	C₄×C₈	C₂×Dic₆₇	1	1	yes	characteristic
183	C₄×C₈	C₂×Dic₆₇	1	1	yes	characteristic
184	C₄×C₈	C₂²×D₆₇	1	1	yes	characteristic
185	C₄×C₈	C₂²×D₆₇	1	4	yes	—
186	C₄×C₈	C₂²×D₆₇	1	1	yes	characteristic
187	C₂×Q₈	C₂×C₆₇⋊C₈	1	2	yes	—
188	C₂×Q₈	C₂×C₆₇⋊C₈	1	2	yes	—
189	C₂²×C₄	C₂×C₆₇⋊C₈	1	1	yes	characteristic
190	C₂²×C₄	C₂×C₆₇⋊C₈	1	1	yes	characteristic
191	C₂²×C₄	C₂²×Dic₆₇	1	1	yes	characteristic
192	M₁₆	—	2	4	no	—
193	C₂×C₈	—	2	2	no	—
194	C₂×C₈	C₂²×Dic₆₇	1	1	yes	characteristic
195	C₂×C₈	C₂²×Dic₆₇	1	1	yes	characteristic
196	C₂×C₈	C₂²×Dic₆₇	1	4	yes	—
197	C₂×C₈	C₂²×Dic₆₇	1	1	yes	characteristic
198	C₂×C₈	Q₈×D₆₇	1	2	yes	—
199	C₂×C₈	#11761629	1	2	yes	—
200	C₂×C₈	C₂³×D₆₇	1	1	yes	characteristic
201	C₄⋊C₄	C₂×C₆₇⋊C₈	1	2	yes	—
202	C₄⋊C₄	C₂×C₆₇⋊C₈	1	2	yes	—
203	C₄⋊C₄	C₂×C₆₇⋊C₈	1	4	yes	—
204	C₄⋊C₄	C₂×C₆₇⋊C₈	1	4	yes	—
205	C₄⋊C₄	C₂×C₆₇⋊C₈	1	2	yes	—
206	C₄⋊C₄	C₂×C₆₇⋊C₈	1	2	yes	—
207	C₂²⋊C₄	C₂×C₆₇⋊C₈	1	4	yes	—
208	C₄²	C₂×C₆₇⋊C₈	1	1	yes	characteristic
209	C₄²	C₂×C₆₇⋊C₈	1	1	yes	characteristic
210	C₄²	C₂²×Dic₆₇	1	1	yes	characteristic
211	C₄²	C₂²×Dic₆₇	1	1	yes	characteristic
212	C₄²	C₂²×Dic₆₇	1	4	yes	—
213	C₁₆	—	67	4	no	—
214	C₁₆	—	134	2	no	—
215	C₁₆	—	134	2	no	—
216	C₁₆	—	134	2	no	—
217	C₂³	C₂²×C₆₇⋊C₈	1	1	yes	characteristic
218	Q₈	—	2	4	no	—
219	Q₈	—	4	2	no	—
220	C₂×C₄	—	2	2	no	—
221	C₂×C₄	—	4	1	no	—
222	C₂×C₄	Q₈×Dic₆₇	1	2	yes	—
223	C₂×C₄	C₂²×C₆₇⋊C₈	1	1	yes	characteristic
224	C₂×C₄	C₂²×C₆₇⋊C₈	1	1	yes	characteristic
225	C₂×C₄	C₂²×C₆₇⋊C₈	1	1	yes	characteristic
226	C₂×C₄	C₂²×C₆₇⋊C₈	1	4	yes	—
227	C₂×C₄	C₂²×C₆₇⋊C₈	1	4	yes	—
228	C₂×C₄	C₂²×C₆₇⋊C₈	1	1	yes	characteristic
229	C₂×C₄	C₂²×C₆₇⋊C₈	1	1	yes	characteristic
230	C₂×C₄	#14524050	1	2	yes	—
231	C₂×C₄	C₂³×Dic₆₇	1	1	yes	characteristic
232	C₈	—	2	1	no	—
233	C₈	—	2	1	no	—
234	C₈	—	2	4	no	—
235	C₈	Q₈×Dic₆₇	1	2	yes	—
236	C₈	C₂×Q₈×D₆₇	1	1	yes	characteristic
237	C₈	C₂×#11761629	1	1	yes	characteristic
238	C₂²	—	2	1	no	—
239	C₂²	—	2	1	no	—
240	C₂²	Q₈×C₆₇⋊C₈	1	2	yes	—
241	C₂²	C₂³×C₆₇⋊C₈	1	1	yes	characteristic
242	C₄	—	2	1	no	—
243	C₄	—	2	1	no	—
244	C₄	—	2	4	no	—
245	C₄	—	2	1	no	—
246	C₄	—	2	4	no	—
247	C₄	Q₈×C₆₇⋊C₈	1	2	yes	—
248	C₄	#16863813	1	2	yes	—
249	C₄	C₂×Q₈×Dic₆₇	1	1	yes	characteristic
250	C₄	C₂×#14524050	1	1	yes	characteristic
251	C₂	—	2	2	no	—
252	C₂	C₂×Q₈×C₆₇⋊C₈	1	1	yes	characteristic
253	C₂	C₂×#16863813	1	1	yes	characteristic
254	C₂	#22573312	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₈	C₂³×D₆₇
commutator, socle	C₂²×C₆₇	C₂³×C₈
Fitting	C₆₇×#2960	C₂
prime radical	C₂³×C₆₇	C₂²×C₈
cosocle	C₂×C₈×C₆₇	C₂⁴
hypercenter	#2960	D₆₇
hypocenter	C₆₇	#33476
LCS	G ▷ C₂²×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂×C₈ ◁ #2960

Complements

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

N	H ≅ G/N	type
1. #22562118	C₂	semidirect
2. #22562118	C₂	semidirect
3. Q₈×C₆₇⋊C₁₆	C₂	semidirect
31. #14523983	Q₈	semidirect
136. C₆₇	#33476	semidirect

Maximal subgroups

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

subgroup	index
#22562118	2
Q₈×C₆₇⋊C₁₆	2
#22561968	2
C₂×C₄⋊(C₆₇⋊C₁₆)	2
#22561806	2
C₄×#14523983	2
C₆₇×#2960	2
#33476	67