Subgroups — #31008493

Order 17152.

716 conjugacy classes of subgroups.

The subgroup lattice has 162 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22573312	C₂	1	2	yes	maximal
2	#22561814	C₂	1	6	yes	maximal, complemented
3	C₂×#16863174	C₂	1	6	yes	maximal, complemented
4	C₆₇×#2875	C₂	1	1	yes	maximal, characteristic
5	#16863813	C₄	1	16	yes	complemented
6	C₂×#14523983	C₂²	1	12	yes	—
7	#16863174	C₂²	1	4	yes	—
8	#16863174	C₂²	1	12	yes	—
9	C₂×#131×C₆₇	C₄	1	1	yes	characteristic
10	C₂×#131×C₆₇	C₂²	1	1	yes	characteristic
11	C₄×#41×C₆₇	C₄	1	1	yes	characteristic
12	#405×C₆₇	C₄	1	3	yes	—
13	#405×C₆₇	C₂²	1	3	yes	—
14	C₄×M₁₆×C₆₇	C₄	1	3	yes	—
15	C₂×#97×C₆₇	C₂²	1	3	yes	—
16	#14523983	C₂×C₄	1	24	yes	complemented
17	#14523983	C₂³	1	8	yes	—
18	C₂×C₆₇⋊C₁₆	C₂×C₄	1	24	yes	complemented
19	C₂×#41×C₆₇	C₂×C₄	1	1	yes	characteristic
20	#131×C₆₇	—	2	4	no	—
21	#131×C₆₇	C₂×C₄	1	8	yes	—
22	C₂×M₁₆×C₆₇	C₂×C₄	1	3	yes	—
23	C₂×M₁₆×C₆₇	C₂×C₄	1	3	yes	—
24	C₂²×C₈×C₆₇	C₂×C₄	1	3	yes	—
25	C₂²×C₈×C₆₇	C₂³	1	3	yes	—
26	C₄×Q₈×C₆₇	—	2	1	no	—
27	C₄×D₄×C₆₇	—	2	3	no	—
28	#117×C₆₇	C₂×C₄	1	3	yes	—
29	C₂×C₄²×C₆₇	C₂×C₄	1	3	yes	—
30	#97×C₆₇	C₂×C₄	1	3	yes	—
31	#97×C₆₇	C₂×C₄	1	3	yes	—
32	#97×C₆₇	C₂³	1	1	yes	characteristic
33	#97×C₆₇	C₂³	1	3	yes	—
34	C₄×C₈×C₆₇	C₂×C₄	1	6	yes	—
35	C₆₇⋊C₁₆	C₂²×C₄	1	16	yes	—
36	#41×C₆₇	C₄²	1	4	yes	—
37	#41×C₆₇	M₁₆	1	4	yes	—
38	C₂×Q₈×C₆₇	M₁₆	1	1	yes	characteristic
39	C₂×D₄×C₆₇	M₁₆	1	3	yes	—
40	C₂²×C₄×C₆₇	—	2	3	no	—
41	C₂²×C₄×C₆₇	C₂²×C₄	1	3	yes	—
42	M₁₆×C₆₇	—	2	6	no	—
43	M₁₆×C₆₇	C₄²	1	12	yes	—
44	C₂×C₈×C₆₇	—	2	6	no	—
45	C₂×C₈×C₆₇	C₂²×C₄	1	12	yes	—
46	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
47	C₂×C₈×C₆₇	C₂²×C₄	1	3	yes	—
48	C₂×C₈×C₆₇	C₂²×C₄	1	3	yes	—
49	C₂×C₈×C₆₇	C₂⁴	1	1	yes	characteristic
50	C₄⋊C₄×C₆₇	—	2	3	no	—
51	C₂²⋊C₄×C₆₇	—	2	3	no	—
52	C₄²×C₆₇	—	2	3	no	—
53	C₄²×C₆₇	C₂²×C₄	1	1	yes	characteristic
54	C₄²×C₆₇	C₂²×C₄	1	3	yes	—
55	C₂³×C₆₇	C₂×M₁₆	1	3	yes	—
56	Q₈×C₆₇	#97	1	4	yes	—
57	D₄×C₆₇	#97	1	12	yes	—
58	C₂×C₄×C₆₇	—	2	1	no	—
59	C₂×C₄×C₆₇	—	2	3	no	—
60	C₂×C₄×C₆₇	—	8	3	no	—
61	C₂×C₄×C₆₇	C₂×C₄²	1	6	yes	—
62	C₂×C₄×C₆₇	C₂×M₁₆	1	6	yes	—
63	C₂×C₄×C₆₇	C₂×M₁₆	1	3	yes	—
64	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
65	C₈×C₆₇	—	2	1	no	—
66	C₈×C₆₇	—	2	3	no	—
67	C₈×C₆₇	C₂×C₄²	1	6	yes	—
68	C₈×C₆₇	C₂³×C₄	1	2	yes	—
69	C₂²×C₆₇	—	4	3	no	—
70	C₂²×C₆₇	C₂×#97	1	6	yes	—
71	C₂²×C₆₇	C₂²×M₁₆	1	1	yes	characteristic
72	C₄×C₆₇	—	4	3	no	—
73	C₄×C₆₇	—	4	1	no	—
74	C₄×C₆₇	C₂×#97	1	6	yes	—
75	C₄×C₆₇	C₂²×C₄²	1	1	yes	characteristic
76	C₄×C₆₇	C₂²×M₁₆	1	1	yes	characteristic
77	#33329	—	67	1	no	maximal, Sylow
78	C₂×C₆₇	—	2	1	no	—
79	C₂×C₆₇	—	2	6	no	—
80	C₂×C₆₇	C₂²×#97	1	1	yes	characteristic
81	#3407	—	67	2	no	—
82	#2875	D₆₇	1	1	yes	characteristic
83	#2110	—	67	6	no	—
84	C₂×#347	—	67	6	no	—
85	C₆₇	#33329	1	1	yes	characteristic, complemented, Sylow
86	C₂×#131	Dic₆₇	1	1	yes	characteristic
87	C₂×#131	C₂×D₆₇	1	1	yes	characteristic
88	C₄×#41	Dic₆₇	1	1	yes	characteristic
89	#504	—	67	16	no	—
90	C₂×M₃₂	—	67	12	no	—
91	#405	Dic₆₇	1	3	yes	—
92	#405	C₂×D₆₇	1	3	yes	—
93	C₄×M₁₆	Dic₆₇	1	3	yes	—
94	C₂×#97	C₂×D₆₇	1	3	yes	—
95	#347	—	67	4	no	—
96	#347	—	67	12	no	—
97	C₂×#41	C₂×Dic₆₇	1	1	yes	characteristic
98	#131	—	2	4	no	—
99	#131	C₄×D₆₇	1	8	yes	—
100	C₂×M₁₆	C₂×Dic₆₇	1	3	yes	—
101	C₂×M₁₆	C₂×Dic₆₇	1	3	yes	—
102	C₂²×C₈	C₂×Dic₆₇	1	3	yes	—
103	C₂²×C₈	C₂²×D₆₇	1	3	yes	—
104	C₄×Q₈	—	2	1	no	—
105	C₄×D₄	—	2	3	no	—
106	#117	C₂×Dic₆₇	1	3	yes	—
107	C₂×C₄²	C₂×Dic₆₇	1	3	yes	—
108	M₃₂	—	67	24	no	—
109	M₃₂	—	67	8	no	—
110	C₂×C₁₆	—	67	24	no	—
111	#97	C₂×Dic₆₇	1	3	yes	—
112	#97	C₂×Dic₆₇	1	3	yes	—
113	#97	C₂²×D₆₇	1	1	yes	characteristic
114	#97	C₂²×D₆₇	1	3	yes	—
115	C₄×C₈	C₂×Dic₆₇	1	6	yes	—
116	#41	C₄×Dic₆₇	1	4	yes	—
117	#41	#11761612	1	4	yes	—
118	C₂×Q₈	#11761612	1	1	yes	characteristic
119	C₂×D₄	#11761612	1	3	yes	—
120	C₂²×C₄	—	2	3	no	—
121	C₂²×C₄	C₂²×Dic₆₇	1	3	yes	—
122	M₁₆	—	2	6	no	—
123	M₁₆	C₄×Dic₆₇	1	12	yes	—
124	C₂×C₈	—	2	6	no	—
125	C₂×C₈	C₂×C₄×D₆₇	1	12	yes	—
126	C₂×C₈	C₂²×Dic₆₇	1	3	yes	—
127	C₂×C₈	C₂²×Dic₆₇	1	1	yes	characteristic
128	C₂×C₈	C₂²×Dic₆₇	1	3	yes	—
129	C₂×C₈	C₂³×D₆₇	1	1	yes	characteristic
130	C₄⋊C₄	—	2	3	no	—
131	C₂²⋊C₄	—	2	3	no	—
132	C₄²	—	2	3	no	—
133	C₄²	C₂²×Dic₆₇	1	3	yes	—
134	C₄²	C₂²×Dic₆₇	1	1	yes	characteristic
135	C₁₆	—	67	16	no	—
136	C₂³	C₂×#11761612	1	3	yes	—
137	Q₈	#14523954	1	4	yes	—
138	D₄	#14523954	1	12	yes	—
139	C₂×C₄	—	2	1	no	—
140	C₂×C₄	—	2	3	no	—
141	C₂×C₄	—	8	3	no	—
142	C₂×C₄	C₂×C₄×Dic₆₇	1	6	yes	—
143	C₂×C₄	C₂×#11761612	1	3	yes	—
144	C₂×C₄	C₂×#11761612	1	6	yes	—
145	C₂×C₄	C₂³×Dic₆₇	1	1	yes	characteristic
146	C₈	—	2	1	no	—
147	C₈	—	2	3	no	—
148	C₈	C₂×C₄×Dic₆₇	1	6	yes	—
149	C₈	C₂²×C₄×D₆₇	1	2	yes	—
150	C₂²	—	4	3	no	—
151	C₂²	C₂×#14523954	1	6	yes	—
152	C₂²	C₂²×#11761612	1	1	yes	characteristic
153	C₄	—	4	1	no	—
154	C₄	—	4	3	no	—
155	C₄	C₂×#14523954	1	6	yes	—
156	C₄	C₂²×C₄×Dic₆₇	1	1	yes	characteristic
157	C₄	C₂²×#11761612	1	1	yes	characteristic
158	C₂	—	2	1	no	—
159	C₂	—	2	6	no	—
160	C₂	C₂²×#14523954	1	1	yes	characteristic

	subgroup	G/H
center	C₄	C₂²×C₄×Dic₆₇
commutator	C₄×C₆₇	C₂²×C₄²
Frattini	C₂×C₈	C₂³×D₆₇
Fitting	C₆₇×#2875	C₂
socle	C₂×C₆₇	C₂²×#97
prime radical	C₂×#41×C₆₇	C₂×C₄
cosocle	C₂×C₈×C₆₇	C₂⁴
hypercenter	#2875	D₆₇
hypocenter	C₆₇	#33329
LCS	G ▷ C₄×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₄ ◁ #2875

Complements

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

N	H ≅ G/N	type
2. #22561814	C₂	semidirect
3. C₂×#16863174	C₂	semidirect
5. #16863813	C₄	semidirect
16. #14523983	C₂×C₄	semidirect
18. C₂×C₆₇⋊C₁₆	C₂×C₄	semidirect
85. C₆₇	#33329	semidirect

Maximal subgroups

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

subgroup	index
#22573312	2
#22561814	2
C₂×#16863174	2
C₆₇×#2875	2
#33329	67