Subgroups — #31008472

Order 17152.

826 conjugacy classes of subgroups.

The subgroup lattice has 196 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#14523983	C₂	1	2	yes	maximal
2	#22561806	C₂	1	8	yes	maximal, complemented
3	C₄×#14523983	C₂	1	4	yes	maximal, complemented
4	C₂×#405×C₆₇	C₂	1	1	yes	maximal, characteristic
5	C₂×#14523983	C₄	1	16	yes	complemented
6	C₂×#14523983	C₂²	1	4	yes	—
7	C₂×#14523983	C₂²	1	4	yes	—
8	C₂²×C₆₇⋊C₁₆	C₂²	1	4	yes	—
9	#16863172	C₂²	1	8	yes	complemented
10	C₄×C₆₇⋊C₁₆	C₂²	1	8	yes	complemented
11	C₂²×M₁₆×C₆₇	C₄	1	1	yes	characteristic
12	C₂³×C₈×C₆₇	C₂²	1	1	yes	characteristic
13	C₂×#117×C₆₇	C₄	1	1	yes	characteristic
14	#405×C₆₇	C₄	1	4	yes	—
15	#405×C₆₇	C₂²	1	4	yes	—
16	C₂×#97×C₆₇	C₄	1	2	yes	—
17	C₂×C₄×C₈×C₆₇	C₂²	1	2	yes	—
18	#14523983	C₂×C₄	1	16	yes	—
19	#14523983	C₂×C₄	1	16	yes	complemented
20	C₂×C₆₇⋊C₁₆	C₂×C₄	1	16	yes	complemented
21	C₂×C₆₇⋊C₁₆	C₂³	1	8	yes	—
22	C₂³×C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
23	C₂×M₁₆×C₆₇	C₈	1	8	yes	—
24	C₂×M₁₆×C₆₇	C₂×C₄	1	4	yes	—
25	C₂²×C₈×C₆₇	C₂×C₄	1	8	yes	—
26	C₂²×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
27	C₂²×C₈×C₆₇	C₂×C₄	1	2	yes	—
28	C₂²×C₈×C₆₇	C₂×C₄	1	2	yes	—
29	C₂²×C₈×C₆₇	C₂³	1	2	yes	—
30	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
31	#117×C₆₇	C₈	1	4	yes	—
32	#117×C₆₇	C₂×C₄	1	4	yes	—
33	C₂×C₄⋊C₄×C₆₇	C₈	1	2	yes	—
34	C₂×C₂²⋊C₄×C₆₇	C₈	1	2	yes	—
35	C₂×C₄²×C₆₇	C₂×C₄	1	2	yes	—
36	#97×C₆₇	C₂×C₄	1	8	yes	—
37	C₄×C₈×C₆₇	C₂×C₄	1	4	yes	—
38	C₄×C₈×C₆₇	C₂³	1	4	yes	—
39	C₆₇⋊C₁₆	C₂²×C₄	1	16	yes	—
40	C₂⁴×C₆₇	C₂×C₈	1	1	yes	characteristic
41	C₂²×C₄×C₆₇	C₄²	1	4	yes	—
42	C₂²×C₄×C₆₇	C₂×C₈	1	4	yes	—
43	C₂²×C₄×C₆₇	C₂×C₈	1	1	yes	characteristic
44	C₂²×C₄×C₆₇	C₂×C₈	1	2	yes	—
45	C₂²×C₄×C₆₇	C₂×C₈	1	4	yes	—
46	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
47	C₂²×C₄×C₆₇	C₂²×C₄	1	2	yes	—
48	M₁₆×C₆₇	C₂×C₈	1	16	yes	—
49	C₂×C₈×C₆₇	C₄²	1	4	yes	—
50	C₂×C₈×C₆₇	C₄²	1	8	yes	—
51	C₂×C₈×C₆₇	C₂×C₈	1	8	yes	—
52	C₂×C₈×C₆₇	C₂²×C₄	1	4	yes	—
53	C₂×C₈×C₆₇	C₂²×C₄	1	4	yes	—
54	C₂×C₈×C₆₇	C₂²×C₄	1	4	yes	—
55	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
56	C₂×C₈×C₆₇	C₂²×C₄	1	2	yes	—
57	C₂×C₈×C₆₇	C₂²×C₄	1	4	yes	—
58	C₂×C₈×C₆₇	C₂⁴	1	1	yes	characteristic
59	C₄⋊C₄×C₆₇	C₂×C₈	1	8	yes	—
60	C₂²⋊C₄×C₆₇	C₂×C₈	1	8	yes	—
61	C₄²×C₆₇	C₂×C₈	1	4	yes	—
62	C₄²×C₆₇	C₂²×C₄	1	4	yes	—
63	C₂³×C₆₇	—	4	2	no	—
64	C₂³×C₆₇	C₄×C₈	1	4	yes	—
65	C₂³×C₆₇	C₂²×C₈	1	2	yes	—
66	C₂³×C₆₇	C₂²×C₈	1	1	yes	characteristic
67	C₂×C₄×C₆₇	—	4	4	no	—
68	C₂×C₄×C₆₇	C₄×C₈	1	4	yes	—
69	C₂×C₄×C₆₇	C₄×C₈	1	8	yes	—
70	C₂×C₄×C₆₇	C₂×C₄²	1	4	yes	—
71	C₂×C₄×C₆₇	C₂×C₄²	1	2	yes	—
72	C₂×C₄×C₆₇	C₂²×C₈	1	8	yes	—
73	C₂×C₄×C₆₇	C₂²×C₈	1	2	yes	—
74	C₂×C₄×C₆₇	C₂²×C₈	1	4	yes	—
75	C₂×C₄×C₆₇	C₂²×C₈	1	1	yes	characteristic
76	C₂×C₄×C₆₇	C₂²×C₈	1	2	yes	—
77	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
78	C₈×C₆₇	C₂×C₄²	1	4	yes	—
79	C₈×C₆₇	C₂×C₄²	1	2	yes	—
80	C₈×C₆₇	C₂²×C₈	1	8	yes	—
81	C₈×C₆₇	C₂³×C₄	1	2	yes	—
82	C₂²×C₆₇	—	2	2	no	—
83	C₂²×C₆₇	—	2	4	no	—
84	C₂²×C₆₇	—	4	4	no	—
85	C₂²×C₆₇	C₂×C₄×C₈	1	2	yes	—
86	C₂²×C₆₇	C₂×C₄×C₈	1	4	yes	—
87	C₂²×C₆₇	C₂³×C₈	1	1	yes	characteristic
88	C₄×C₆₇	—	2	8	no	—
89	C₄×C₆₇	C₂×C₄×C₈	1	2	yes	—
90	C₄×C₆₇	C₂×C₄×C₈	1	4	yes	—
91	C₄×C₆₇	C₂²×C₄²	1	1	yes	characteristic
92	C₄×C₆₇	C₂³×C₈	1	1	yes	characteristic
93	#33324	—	67	1	no	maximal, Sylow
94	C₂×C₆₇	—	2	2	no	—
95	C₂×C₆₇	—	2	4	no	—
96	C₂×C₆₇	C₂²×C₄×C₈	1	1	yes	characteristic
97	C₂×C₆₇	#3407	1	2	yes	—
98	C₂²×M₃₂	—	67	2	no	—
99	C₂×#405	D₆₇	1	1	yes	characteristic
100	#2108	—	67	8	no	—
101	C₄×M₃₂	—	67	4	no	—
102	C₆₇	#33324	1	1	yes	characteristic, complemented, Sylow
103	C₂²×M₁₆	Dic₆₇	1	1	yes	characteristic
104	C₂³×C₈	C₂×D₆₇	1	1	yes	characteristic
105	C₂×#117	Dic₆₇	1	1	yes	characteristic
106	C₂×M₃₂	—	67	16	no	—
107	C₂×M₃₂	—	67	4	no	—
108	C₂×M₃₂	—	67	4	no	—
109	C₂²×C₁₆	—	67	4	no	—
110	#405	Dic₆₇	1	4	yes	—
111	#405	C₂×D₆₇	1	4	yes	—
112	C₂×#97	Dic₆₇	1	2	yes	—
113	C₂×C₄×C₈	C₂×D₆₇	1	2	yes	—
114	#346	—	67	8	no	—
115	C₄×C₁₆	—	67	8	no	—
116	C₂³×C₄	C₂×Dic₆₇	1	1	yes	characteristic
117	C₂×M₁₆	C₆₇⋊C₈	1	8	yes	—
118	C₂×M₁₆	C₂×Dic₆₇	1	4	yes	—
119	C₂²×C₈	C₄×D₆₇	1	8	yes	—
120	C₂²×C₈	C₂×Dic₆₇	1	1	yes	characteristic
121	C₂²×C₈	C₂×Dic₆₇	1	2	yes	—
122	C₂²×C₈	C₂×Dic₆₇	1	2	yes	—
123	C₂²×C₈	C₂²×D₆₇	1	1	yes	characteristic
124	C₂²×C₈	C₂²×D₆₇	1	2	yes	—
125	#117	C₆₇⋊C₈	1	4	yes	—
126	#117	C₂×Dic₆₇	1	4	yes	—
127	C₂×C₄⋊C₄	C₆₇⋊C₈	1	2	yes	—
128	C₂×C₂²⋊C₄	C₆₇⋊C₈	1	2	yes	—
129	C₂×C₄²	C₂×Dic₆₇	1	2	yes	—
130	M₃₂	—	67	16	no	—
131	M₃₂	—	67	16	no	—
132	C₂×C₁₆	—	67	16	no	—
133	C₂×C₁₆	—	67	8	no	—
134	#97	C₂×Dic₆₇	1	8	yes	—
135	C₄×C₈	C₂×Dic₆₇	1	4	yes	—
136	C₄×C₈	C₂²×D₆₇	1	4	yes	—
137	C₂⁴	C₂×C₆₇⋊C₈	1	1	yes	characteristic
138	C₂²×C₄	C₄×Dic₆₇	1	4	yes	—
139	C₂²×C₄	C₂×C₆₇⋊C₈	1	4	yes	—
140	C₂²×C₄	C₂×C₆₇⋊C₈	1	1	yes	characteristic
141	C₂²×C₄	C₂×C₆₇⋊C₈	1	2	yes	—
142	C₂²×C₄	C₂×C₆₇⋊C₈	1	4	yes	—
143	C₂²×C₄	C₂²×Dic₆₇	1	1	yes	characteristic
144	C₂²×C₄	C₂²×Dic₆₇	1	2	yes	—
145	M₁₆	C₂×C₆₇⋊C₈	1	16	yes	—
146	C₂×C₈	C₄×Dic₆₇	1	4	yes	—
147	C₂×C₈	C₄×Dic₆₇	1	8	yes	—
148	C₂×C₈	C₂×C₆₇⋊C₈	1	8	yes	—
149	C₂×C₈	C₂×C₄×D₆₇	1	4	yes	—
150	C₂×C₈	C₂×C₄×D₆₇	1	4	yes	—
151	C₂×C₈	C₂×C₄×D₆₇	1	4	yes	—
152	C₂×C₈	C₂²×Dic₆₇	1	1	yes	characteristic
153	C₂×C₈	C₂²×Dic₆₇	1	2	yes	—
154	C₂×C₈	C₂²×Dic₆₇	1	4	yes	—
155	C₂×C₈	C₂³×D₆₇	1	1	yes	characteristic
156	C₄⋊C₄	C₂×C₆₇⋊C₈	1	8	yes	—
157	C₂²⋊C₄	C₂×C₆₇⋊C₈	1	8	yes	—
158	C₄²	C₂×C₆₇⋊C₈	1	4	yes	—
159	C₄²	C₂²×Dic₆₇	1	4	yes	—
160	C₁₆	—	67	16	no	—
161	C₂³	—	4	2	no	—
162	C₂³	C₄×C₆₇⋊C₈	1	4	yes	—
163	C₂³	C₂²×C₆₇⋊C₈	1	2	yes	—
164	C₂³	C₂²×C₆₇⋊C₈	1	1	yes	characteristic
165	C₂×C₄	—	4	4	no	—
166	C₂×C₄	C₄×C₆₇⋊C₈	1	4	yes	—
167	C₂×C₄	C₄×C₆₇⋊C₈	1	8	yes	—
168	C₂×C₄	C₂×C₄×Dic₆₇	1	4	yes	—
169	C₂×C₄	C₂×C₄×Dic₆₇	1	2	yes	—
170	C₂×C₄	C₂²×C₆₇⋊C₈	1	8	yes	—
171	C₂×C₄	C₂²×C₆₇⋊C₈	1	1	yes	characteristic
172	C₂×C₄	C₂²×C₆₇⋊C₈	1	2	yes	—
173	C₂×C₄	C₂²×C₆₇⋊C₈	1	2	yes	—
174	C₂×C₄	C₂²×C₆₇⋊C₈	1	4	yes	—
175	C₂×C₄	C₂³×Dic₆₇	1	1	yes	characteristic
176	C₈	C₂×C₄×Dic₆₇	1	4	yes	—
177	C₈	C₂×C₄×Dic₆₇	1	2	yes	—
178	C₈	C₂²×C₆₇⋊C₈	1	8	yes	—
179	C₈	C₂²×C₄×D₆₇	1	2	yes	—
180	C₂²	—	2	2	no	—
181	C₂²	—	2	4	no	—
182	C₂²	—	4	4	no	—
183	C₂²	C₂×C₄×C₆₇⋊C₈	1	2	yes	—
184	C₂²	C₂×C₄×C₆₇⋊C₈	1	4	yes	—
185	C₂²	C₂³×C₆₇⋊C₈	1	1	yes	characteristic
186	C₄	—	2	8	no	—
187	C₄	C₂×C₄×C₆₇⋊C₈	1	2	yes	—
188	C₄	C₂×C₄×C₆₇⋊C₈	1	4	yes	—
189	C₄	C₂²×C₄×Dic₆₇	1	1	yes	characteristic
190	C₄	C₂³×C₆₇⋊C₈	1	1	yes	characteristic
191	C₂	—	2	2	no	—
192	C₂	—	2	4	no	—
193	C₂	C₂²×C₄×C₆₇⋊C₈	1	1	yes	characteristic
194	C₂	#22573312	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂×C₈	C₂³×D₆₇
commutator	C₂×C₆₇	C₂²×C₄×C₈
Fitting	C₂×#405×C₆₇	C₂
socle	C₂²×C₆₇	C₂³×C₈
prime radical	C₂⁴×C₆₇	C₂×C₈
cosocle	C₂×C₈×C₆₇	C₂⁴
hypercenter	C₂×#405	D₆₇
hypocenter	C₆₇	#33324
LCS	G ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂×C₈ ◁ C₂×#405

Complements

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

N	H ≅ G/N	type
2. #22561806	C₂	semidirect
3. C₄×#14523983	C₂	semidirect
5. C₂×#14523983	C₄	semidirect
9. #16863172	C₂²	semidirect
10. C₄×C₆₇⋊C₁₆	C₂²	semidirect
19. #14523983	C₂×C₄	semidirect
20. C₂×C₆₇⋊C₁₆	C₂×C₄	semidirect
102. C₆₇	#33324	semidirect

Maximal subgroups

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

subgroup	index
C₂²×#14523983	2
#22561806	2
C₄×#14523983	2
C₂×#405×C₆₇	2
#33324	67