Subgroups — #30872606

Order 17152.

970 conjugacy classes of subgroups.

The subgroup lattice has 232 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	Q₈×#11761612	C₂	1	2	yes	maximal
2	#22561054	C₂	1	6	yes	maximal
3	#22560205	C₂	1	3	yes	maximal
4	C₄×#14523962	C₂	1	3	yes	maximal
5	C₂×#557×C₆₇	C₂	1	1	yes	maximal, characteristic
6	#16863527	—	2	6	no	—
7	Q₈×C₆₇⋊C₈	—	2	2	no	—
8	#16863420	C₂²	1	6	yes	—
9	C₂×#14523962	C₂²	1	6	yes	—
10	C₂×#14523962	C₂²	1	6	yes	—
11	C₄×#11761612	C₂²	1	6	yes	—
12	#16863165	C₂²	1	1	yes	characteristic
13	#16863165	C₂²	1	3	yes	—
14	#557×C₆₇	—	2	2	no	—
15	#557×C₆₇	—	2	6	no	—
16	C₂×C₄⋊Q₈×C₆₇	C₂²	1	3	yes	—
17	C₂×#125×C₆₇	C₄	1	6	yes	—
18	C₂×C₄×Q₈×C₆₇	C₄	1	2	yes	—
19	C₂×C₄×Q₈×C₆₇	C₂²	1	3	yes	—
20	C₂×C₄×Q₈×C₆₇	C₂²	1	1	yes	characteristic
21	C₂×#11761612	C₂³	1	6	yes	—
22	C₂×#11761612	C₂³	1	2	yes	—
23	C₄⋊(C₆₇⋊C₈)	—	2	6	no	—
24	C₄⋊(C₆₇⋊C₈)	—	2	6	no	—
25	#14523962	—	2	12	no	—
26	#14523962	—	4	6	no	—
27	#14523954	—	2	6	no	—
28	C₄×C₆₇⋊C₈	—	2	6	no	—
29	C₂²×Q₈×C₆₇	C₂×C₄	1	1	yes	characteristic
30	C₂²×Q₈×C₆₇	C₂³	1	3	yes	—
31	C₄⋊Q₈×C₆₇	—	2	6	no	—
32	C₄⋊Q₈×C₆₇	—	2	6	no	—
33	#125×C₆₇	—	2	12	no	—
34	#125×C₆₇	—	2	12	no	—
35	C₄×Q₈×C₆₇	—	2	6	no	—
36	C₄×Q₈×C₆₇	—	2	2	no	—
37	C₄×Q₈×C₆₇	—	2	6	no	—
38	C₄×Q₈×C₆₇	—	2	6	no	—
39	C₄×Q₈×C₆₇	D₄	1	2	yes	—
40	C₄×Q₈×C₆₇	D₄	1	6	yes	—
41	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	6	yes	—
42	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	6	yes	—
43	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	6	yes	—
44	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	3	yes	—
45	C₂×C₄⋊C₄×C₆₇	C₂³	1	1	yes	characteristic
46	C₂×C₄²×C₆₇	C₂×C₄	1	6	yes	—
47	C₂×C₄²×C₆₇	C₂³	1	3	yes	—
48	#11761612	—	2	2	no	—
49	#11761612	—	2	6	no	—
50	#11761612	—	2	6	no	—
51	#11761612	#41	1	4	yes	—
52	C₂×C₆₇⋊C₈	—	2	2	no	—
53	C₂×C₆₇⋊C₈	—	2	6	no	—
54	C₂×Q₈×C₆₇	—	2	4	no	—
55	C₂×Q₈×C₆₇	—	2	6	no	—
56	C₂×Q₈×C₆₇	—	2	6	no	—
57	C₂×Q₈×C₆₇	—	2	3	no	—
58	C₂×Q₈×C₆₇	—	2	3	no	—
59	C₂×Q₈×C₆₇	C₂²⋊C₄	1	3	yes	—
60	C₂×Q₈×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
61	C₂²×C₄×C₆₇	C₂²×C₄	1	6	yes	—
62	C₂²×C₄×C₆₇	C₂²×C₄	1	3	yes	—
63	C₂²×C₄×C₆₇	C₂²×C₄	1	2	yes	—
64	C₂²×C₄×C₆₇	C₂²×C₄	1	3	yes	—
65	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
66	C₄⋊C₄×C₆₇	—	2	12	no	—
67	C₄⋊C₄×C₆₇	—	2	6	no	—
68	C₄⋊C₄×C₆₇	—	2	6	no	—
69	C₄⋊C₄×C₆₇	—	2	12	no	—
70	C₄⋊C₄×C₆₇	—	2	2	no	—
71	C₄⋊C₄×C₆₇	C₂²⋊C₄	1	6	yes	—
72	C₄⋊C₄×C₆₇	C₂²⋊C₄	1	6	yes	—
73	C₄²×C₆₇	—	2	6	no	—
74	C₄²×C₆₇	—	2	6	no	—
75	C₄²×C₆₇	C₂×D₄	1	6	yes	—
76	C₄²×C₆₇	C₂×D₄	1	6	yes	—
77	C₂³×C₆₇	C₂³×C₄	1	1	yes	characteristic
78	Q₈×C₆₇	—	2	6	no	—
79	Q₈×C₆₇	—	4	6	no	—
80	Q₈×C₆₇	—	4	6	no	—
81	Q₈×C₆₇	#101	1	4	yes	complemented
82	C₂×C₄×C₆₇	—	2	6	no	—
83	C₂×C₄×C₆₇	—	2	2	no	—
84	C₂×C₄×C₆₇	—	2	6	no	—
85	C₂×C₄×C₆₇	—	2	6	no	—
86	C₂×C₄×C₆₇	—	2	6	no	—
87	C₂×C₄×C₆₇	—	4	3	no	—
88	C₂×C₄×C₆₇	—	4	2	no	—
89	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	3	yes	—
90	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	6	yes	—
91	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	3	yes	—
92	C₂×C₄×C₆₇	C₂²×D₄	1	2	yes	—
93	C₂×C₄×C₆₇	C₂×#41	1	2	yes	—
94	C₂×C₄×C₆₇	H(5,𝔽₂)	1	1	yes	characteristic
95	C₂×C₄×C₆₇	2₋¹⁺⁴	1	1	yes	characteristic
96	C₆₇⋊C₈	—	2	4	no	—
97	C₆₇⋊C₈	—	4	6	no	—
98	C₂²×C₆₇	—	2	1	no	—
99	C₂²×C₆₇	—	2	1	no	—
100	C₂²×C₆₇	C₂²×C₂²⋊C₄	1	1	yes	characteristic
101	C₂²×C₆₇	C₄×#41	1	1	yes	characteristic
102	C₂²×C₆₇	#520	1	1	yes	characteristic
103	C₄×C₆₇	—	2	3	no	—
104	C₄×C₆₇	—	2	6	no	—
105	C₄×C₆₇	—	4	6	no	—
106	C₄×C₆₇	—	4	2	no	—
107	C₄×C₆₇	C₂×#101	1	6	yes	—
108	C₄×C₆₇	#548	1	2	yes	—
109	C₄×C₆₇	#550	1	2	yes	—
110	#22538	—	67	1	no	maximal, Sylow
111	C₂×C₆₇	—	2	1	no	—
112	C₂×C₆₇	—	2	1	no	—
113	C₂×C₆₇	#2341	1	1	yes	characteristic
114	C₂×C₆₇	C₂²×#101	1	1	yes	characteristic
115	C₂×C₆₇	#2889	1	1	yes	characteristic
116	C₂×#557	D₆₇	1	1	yes	characteristic
117	Q₈×M₁₆	—	67	2	no	—
118	#1966	—	67	6	no	—
119	#1801	—	67	3	no	—
120	C₄×#101	—	67	3	no	—
121	C₆₇	#22538	1	1	yes	characteristic, complemented, Sylow
122	#557	—	2	2	no	—
123	#557	—	2	6	no	—
124	C₂×C₄⋊Q₈	C₂×D₆₇	1	3	yes	—
125	C₂×#125	Dic₆₇	1	6	yes	—
126	C₂×C₄×Q₈	Dic₆₇	1	2	yes	—
127	C₂×C₄×Q₈	C₂×D₆₇	1	3	yes	—
128	C₂×C₄×Q₈	C₂×D₆₇	1	1	yes	characteristic
129	#446	—	134	6	no	—
130	C₈×Q₈	—	134	2	no	—
131	#423	—	67	6	no	—
132	C₂×#101	—	67	6	no	—
133	C₂×#101	—	67	6	no	—
134	C₄×M₁₆	—	67	6	no	—
135	#343	—	67	3	no	—
136	#343	—	67	1	no	—
137	C₂²×Q₈	C₂×Dic₆₇	1	1	yes	characteristic
138	C₂²×Q₈	C₂²×D₆₇	1	3	yes	—
139	C₂×M₁₆	—	67	2	no	—
140	C₂×M₁₆	—	67	6	no	—
141	C₄⋊Q₈	—	2	6	no	—
142	C₄⋊Q₈	—	2	6	no	—
143	#125	—	2	12	no	—
144	#125	—	2	12	no	—
145	C₄×Q₈	—	2	6	no	—
146	C₄×Q₈	—	2	2	no	—
147	C₄×Q₈	—	2	6	no	—
148	C₄×Q₈	—	2	6	no	—
149	C₄×Q₈	C₂²⋊D₆₇	1	2	yes	—
150	C₄×Q₈	C₂²⋊D₆₇	1	6	yes	—
151	C₂×C₄⋊C₄	C₂×Dic₆₇	1	6	yes	—
152	C₂×C₄⋊C₄	C₂×Dic₆₇	1	6	yes	—
153	C₂×C₄⋊C₄	C₂×Dic₆₇	1	6	yes	—
154	C₂×C₄⋊C₄	C₂×Dic₆₇	1	3	yes	—
155	C₂×C₄⋊C₄	C₂²×D₆₇	1	1	yes	characteristic
156	C₂×C₄²	C₂×Dic₆₇	1	6	yes	—
157	C₂×C₄²	C₂²×D₆₇	1	3	yes	—
158	C₄⋊C₈	—	134	6	no	—
159	C₄⋊C₈	—	134	6	no	—
160	#101	—	268	6	no	—
161	#101	—	134	12	no	—
162	#97	—	134	6	no	—
163	C₄×C₈	—	134	6	no	—
164	C₂×Q₈	—	2	4	no	—
165	C₂×Q₈	—	2	6	no	—
166	C₂×Q₈	—	2	6	no	—
167	C₂×Q₈	—	2	3	no	—
168	C₂×Q₈	—	2	3	no	—
169	C₂×Q₈	C₂²⋊Dic₆₇	1	3	yes	—
170	C₂×Q₈	C₂²⋊Dic₆₇	1	1	yes	characteristic
171	C₂²×C₄	C₂²×Dic₆₇	1	6	yes	—
172	C₂²×C₄	C₂²×Dic₆₇	1	3	yes	—
173	C₂²×C₄	C₂²×Dic₆₇	1	2	yes	—
174	C₂²×C₄	C₂²×Dic₆₇	1	3	yes	—
175	C₂²×C₄	C₂³×D₆₇	1	1	yes	characteristic
176	M₁₆	—	67	4	no	—
177	M₁₆	—	134	6	no	—
178	M₁₆	—	134	6	no	—
179	M₁₆	—	134	2	no	—
180	C₂×C₈	—	134	6	no	—
181	C₂×C₈	—	134	2	no	—
182	C₄⋊C₄	—	2	12	no	—
183	C₄⋊C₄	—	2	6	no	—
184	C₄⋊C₄	—	2	6	no	—
185	C₄⋊C₄	—	2	12	no	—
186	C₄⋊C₄	—	2	2	no	—
187	C₄⋊C₄	C₂²⋊Dic₆₇	1	6	yes	—
188	C₄⋊C₄	C₂²⋊Dic₆₇	1	6	yes	—
189	C₄²	—	2	6	no	—
190	C₄²	—	2	6	no	—
191	C₄²	C₂×C₂²⋊D₆₇	1	6	yes	—
192	C₄²	C₂×C₂²⋊D₆₇	1	6	yes	—
193	C₂³	C₂³×Dic₆₇	1	1	yes	characteristic
194	Q₈	—	2	6	no	—
195	Q₈	—	4	6	no	—
196	Q₈	—	4	6	no	—
197	Q₈	#14523962	1	4	yes	complemented
198	C₂×C₄	—	2	6	no	—
199	C₂×C₄	—	2	2	no	—
200	C₂×C₄	—	2	6	no	—
201	C₂×C₄	—	2	6	no	—
202	C₂×C₄	—	2	6	no	—
203	C₂×C₄	—	4	3	no	—
204	C₂×C₄	—	4	2	no	—
205	C₂×C₄	C₂×C₂²⋊Dic₆₇	1	3	yes	—
206	C₂×C₄	C₂×C₂²⋊Dic₆₇	1	6	yes	—
207	C₂×C₄	C₂×C₂²⋊Dic₆₇	1	3	yes	—
208	C₂×C₄	C₂²×C₂²⋊D₆₇	1	2	yes	—
209	C₂×C₄	#41×D₆₇	1	2	yes	—
210	C₂×C₄	#14524089	1	1	yes	characteristic
211	C₂×C₄	#14524090	1	1	yes	characteristic
212	C₈	—	268	6	no	—
213	C₈	—	134	4	no	—
214	C₂²	—	2	1	no	—
215	C₂²	—	2	1	no	—
216	C₂²	C₂²×C₂²⋊Dic₆₇	1	1	yes	characteristic
217	C₂²	#41×Dic₆₇	1	1	yes	characteristic
218	C₂²	#16863879	1	1	yes	characteristic
219	C₄	—	2	3	no	—
220	C₄	—	2	6	no	—
221	C₄	—	4	6	no	—
222	C₄	—	4	2	no	—
223	C₄	C₂×#14523962	1	6	yes	—
224	C₄	#16864053	1	2	yes	—
225	C₄	#16864065	1	2	yes	—
226	C₂	—	2	1	no	—
227	C₂	—	2	1	no	—
228	C₂	#22563050	1	1	yes	characteristic
229	C₂	C₂²×#14523962	1	1	yes	characteristic
230	C₂	#22568692	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂²×C₂²⋊Dic₆₇
commutator, prime radical	C₂³×C₆₇	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₆₇
Fitting	C₂×#557×C₆₇	C₂
socle	C₂²×C₆₇	C₂²×C₂²⋊C₄
cosocle	C₂²×C₄×C₆₇	C₂⁴
hypercenter	C₂×#557	D₆₇
hypocenter	C₆₇	#22538
LCS	G ▷ C₂³×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂² ◁ C₂×C₄×Q₈ ◁ C₂×#557

Complements

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

N	H ≅ G/N	type
81. Q₈×C₆₇	#101	semidirect
121. C₆₇	#22538	semidirect
197. Q₈	#14523962	semidirect

Maximal subgroups

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

subgroup	index
Q₈×#11761612	2
#22561054	2
#22560205	2
C₄×#14523962	2
C₂×#557×C₆₇	2
#22538	67