Subgroups — #31008491

Order 17152.

716 conjugacy classes of subgroups.

The subgroup lattice has 182 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22573312	C₂	1	1	yes	maximal, characteristic
2	#22568635	C₂	1	1	yes	maximal, characteristic
3	#22561813	C₂	1	6	yes	maximal, complemented
4	C₂×#16863173	C₂	1	6	yes	maximal, complemented
5	C₆₇×#3407	C₂	1	1	yes	maximal, characteristic
6	C₂×#14524050	C₄	1	1	yes	characteristic
7	#41×Dic₆₇	C₄	1	1	yes	characteristic
8	#16863813	C₄	1	8	yes	complemented
9	C₂×#14523983	C₂²	1	6	yes	—
10	#16863354	C₄	1	3	yes	—
11	#16863351	C₂²	1	3	yes	—
12	M₁₆×Dic₆₇	C₄	1	3	yes	—
13	C₂×#14523953	C₂²	1	3	yes	—
14	#16863173	C₂²	1	4	yes	—
15	#16863173	C₂²	1	12	yes	—
16	C₂×#131×C₆₇	C₂²	1	1	yes	characteristic
17	#504×C₆₇	C₄	1	8	yes	complemented
18	C₂×M₃₂×C₆₇	C₂²	1	6	yes	—
19	#14524050	—	2	4	no	—
20	C₂×#11761612	C₂×C₄	1	3	yes	—
21	C₂²×C₆₇⋊C₈	C₂×C₄	1	3	yes	—
22	Q₈×Dic₆₇	—	2	1	no	—
23	D₄×Dic₆₇	—	2	3	no	—
24	#14524002	C₂×C₄	1	3	yes	—
25	C₂×C₄×Dic₆₇	C₂×C₄	1	3	yes	—
26	#14523983	C₂×C₄	1	12	yes	complemented
27	#14523983	C₂³	1	4	yes	—
28	C₂×C₆₇⋊C₁₆	C₂×C₄	1	12	yes	complemented
29	#14523953	C₂×C₄	1	3	yes	—
30	#14523953	C₂×C₄	1	3	yes	—
31	#14523953	C₂³	1	1	yes	characteristic
32	#14523953	C₂³	1	3	yes	—
33	C₈×Dic₆₇	C₂×C₄	1	6	yes	—
34	C₂×#41×C₆₇	C₂×C₄	1	1	yes	characteristic
35	#131×C₆₇	C₂×C₄	1	4	yes	—
36	#131×C₆₇	C₂×C₄	1	4	yes	—
37	C₂×M₁₆×C₆₇	C₂×C₄	1	3	yes	—
38	C₂²×C₈×C₆₇	C₂³	1	3	yes	—
39	M₃₂×C₆₇	C₂×C₄	1	12	yes	complemented
40	M₃₂×C₆₇	C₂³	1	4	yes	—
41	C₂×C₁₆×C₆₇	C₂×C₄	1	12	yes	complemented
42	C₂²×Dic₆₇	—	2	3	no	—
43	#11761612	—	2	6	no	—
44	C₂×C₆₇⋊C₈	—	2	6	no	—
45	C₂×C₆₇⋊C₈	C₂²×C₄	1	3	yes	—
46	C₂×C₆₇⋊C₈	C₂²×C₄	1	1	yes	characteristic
47	C₄⋊Dic₆₇	—	2	3	no	—
48	C₂²⋊Dic₆₇	—	2	3	no	—
49	C₄×Dic₆₇	—	2	3	no	—
50	C₄×Dic₆₇	C₂²×C₄	1	3	yes	—
51	C₄×Dic₆₇	C₂²×C₄	1	1	yes	characteristic
52	C₆₇⋊C₁₆	C₂²×C₄	1	8	yes	—
53	#41×C₆₇	C₄²	1	4	yes	—
54	#41×C₆₇	M₁₆	1	4	yes	—
55	C₂×Q₈×C₆₇	M₁₆	1	1	yes	characteristic
56	C₂×D₄×C₆₇	M₁₆	1	3	yes	—
57	C₂²×C₄×C₆₇	C₂²×C₄	1	3	yes	—
58	M₁₆×C₆₇	C₄²	1	12	yes	—
59	C₂×C₈×C₆₇	C₂²×C₄	1	3	yes	—
60	C₂×C₈×C₆₇	C₂²×C₄	1	6	yes	—
61	C₂×C₈×C₆₇	C₂²×C₄	1	6	yes	—
62	C₂×C₈×C₆₇	C₂⁴	1	1	yes	characteristic
63	C₁₆×C₆₇	C₂²×C₄	1	8	yes	—
64	C₂³×C₆₇	C₂×M₁₆	1	3	yes	—
65	Q₈×C₆₇	#97	1	4	yes	—
66	D₄×C₆₇	#97	1	12	yes	—
67	C₂×C₄×C₆₇	C₂×C₄²	1	6	yes	—
68	C₂×C₄×C₆₇	C₂×M₁₆	1	6	yes	—
69	C₂×C₄×C₆₇	C₂×M₁₆	1	3	yes	—
70	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
71	C₂×Dic₆₇	—	2	1	no	—
72	C₂×Dic₆₇	—	2	3	no	—
73	C₂×Dic₆₇	—	8	3	no	—
74	C₈×C₆₇	C₂×C₄²	1	6	yes	—
75	C₈×C₆₇	C₂³×C₄	1	1	yes	characteristic
76	C₈×C₆₇	C₂³×C₄	1	1	yes	characteristic
77	C₆₇⋊C₈	—	2	1	no	—
78	C₆₇⋊C₈	—	2	3	no	—
79	C₂²×C₆₇	—	4	3	no	—
80	C₂²×C₆₇	C₂×#97	1	6	yes	—
81	C₂²×C₆₇	C₂²×M₁₆	1	1	yes	characteristic
82	C₄×C₆₇	C₂×#97	1	6	yes	—
83	C₄×C₆₇	C₂²×C₄²	1	1	yes	characteristic
84	C₄×C₆₇	C₂²×M₁₆	1	1	yes	characteristic
85	Dic₆₇	—	4	3	no	—
86	Dic₆₇	—	4	1	no	—
87	#33329	—	67	1	no	maximal, Sylow
88	C₂×C₆₇	—	2	1	no	—
89	C₂×C₆₇	—	2	6	no	—
90	C₂×C₆₇	C₂²×#97	1	1	yes	characteristic
91	#3407	—	67	1	no	—
92	#3407	D₆₇	1	1	yes	characteristic
93	#2875	—	67	1	no	—
94	#2110	—	67	6	no	—
95	C₂×#347	—	67	6	no	—
96	C₆₇	#33329	1	1	yes	characteristic, complemented, Sylow
97	C₂×#131	—	67	1	no	—
98	C₂×#131	C₂×D₆₇	1	1	yes	characteristic
99	C₄×#41	—	67	1	no	—
100	#504	—	67	8	no	—
101	#504	Dic₆₇	1	8	yes	complemented
102	C₂×M₃₂	—	67	6	no	—
103	C₂×M₃₂	C₂×D₆₇	1	6	yes	—
104	#405	—	67	3	no	—
105	#405	—	67	3	no	—
106	C₄×M₁₆	—	67	3	no	—
107	C₂×#97	—	67	3	no	—
108	#347	—	67	4	no	—
109	#347	—	67	12	no	—
110	C₂×#41	C₄×D₆₇	1	1	yes	characteristic
111	#131	—	134	4	no	—
112	#131	C₄×D₆₇	1	4	yes	—
113	#131	C₂×Dic₆₇	1	4	yes	—
114	C₂×M₁₆	—	67	3	no	—
115	C₂×M₁₆	C₄×D₆₇	1	3	yes	—
116	C₂²×C₈	—	67	3	no	—
117	C₂²×C₈	C₂²×D₆₇	1	3	yes	—
118	C₄×Q₈	—	134	1	no	—
119	C₄×D₄	—	134	3	no	—
120	#117	—	67	3	no	—
121	C₂×C₄²	—	67	3	no	—
122	M₃₂	—	67	12	no	—
123	M₃₂	—	67	4	no	—
124	M₃₂	C₂×Dic₆₇	1	12	yes	complemented
125	M₃₂	C₂²×D₆₇	1	4	yes	—
126	C₂×C₁₆	—	67	12	no	—
127	C₂×C₁₆	C₂×Dic₆₇	1	12	yes	complemented
128	#97	—	67	3	no	—
129	#97	—	67	3	no	—
130	#97	—	67	1	no	—
131	#97	—	67	3	no	—
132	C₄×C₈	—	67	6	no	—
133	#41	C₄×Dic₆₇	1	4	yes	—
134	#41	#11761611	1	4	yes	—
135	C₂×Q₈	#11761611	1	1	yes	characteristic
136	C₂×D₄	#11761611	1	3	yes	—
137	C₂²×C₄	—	134	3	no	—
138	C₂²×C₄	C₂×C₄×D₆₇	1	3	yes	—
139	M₁₆	—	134	6	no	—
140	M₁₆	C₄×Dic₆₇	1	12	yes	—
141	C₂×C₈	—	67	3	no	—
142	C₂×C₈	—	67	1	no	—
143	C₂×C₈	—	134	6	no	—
144	C₂×C₈	C₂×C₄×D₆₇	1	3	yes	—
145	C₂×C₈	C₂×C₄×D₆₇	1	6	yes	—
146	C₂×C₈	C₂²×Dic₆₇	1	6	yes	—
147	C₂×C₈	C₂³×D₆₇	1	1	yes	characteristic
148	C₄⋊C₄	—	134	3	no	—
149	C₂²⋊C₄	—	134	3	no	—
150	C₄²	—	67	3	no	—
151	C₄²	—	67	1	no	—
152	C₄²	—	134	3	no	—
153	C₁₆	—	67	8	no	—
154	C₁₆	C₂²×Dic₆₇	1	8	yes	—
155	C₂³	C₂×#11761611	1	3	yes	—
156	Q₈	#14523953	1	4	yes	—
157	D₄	#14523953	1	12	yes	—
158	C₂×C₄	—	536	3	no	—
159	C₂×C₄	—	134	1	no	—
160	C₂×C₄	—	134	3	no	—
161	C₂×C₄	C₂×C₄×Dic₆₇	1	6	yes	—
162	C₂×C₄	C₂×#11761611	1	3	yes	—
163	C₂×C₄	C₂×#11761611	1	6	yes	—
164	C₂×C₄	C₂²×C₄×D₆₇	1	1	yes	characteristic
165	C₈	—	134	1	no	—
166	C₈	—	134	3	no	—
167	C₈	C₂×C₄×Dic₆₇	1	6	yes	—
168	C₈	C₂²×C₄×D₆₇	1	1	yes	characteristic
169	C₈	C₂³×Dic₆₇	1	1	yes	characteristic
170	C₂²	—	4	3	no	—
171	C₂²	C₂×#14523953	1	6	yes	—
172	C₂²	C₂²×#11761611	1	1	yes	characteristic
173	C₄	—	268	1	no	—
174	C₄	—	268	3	no	—
175	C₄	C₂×#14523953	1	6	yes	—
176	C₄	C₂²×C₄×Dic₆₇	1	1	yes	characteristic
177	C₄	C₂²×#11761611	1	1	yes	characteristic
178	C₂	—	2	1	no	—
179	C₂	—	2	6	no	—
180	C₂	C₂²×#14523953	1	1	yes	characteristic

	subgroup	G/H
center	C₄	C₂²×C₄×Dic₆₇
commutator	C₄×C₆₇	C₂²×C₄²
Frattini	C₂×C₈	C₂³×D₆₇
Fitting	C₆₇×#3407	C₂
socle	C₂×C₆₇	C₂²×#97
prime radical	C₂×#41×C₆₇	C₂×C₄
cosocle	C₂×C₈×C₆₇	C₂⁴
hypercenter	#3407	D₆₇
hypocenter	C₆₇	#33329
LCS	G ▷ C₄×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₄ ◁ #3407

Complements

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

N	H ≅ G/N	type
3. #22561813	C₂	semidirect
4. C₂×#16863173	C₂	semidirect
8. #16863813	C₄	semidirect
17. #504×C₆₇	C₄	semidirect
26. #14523983	C₂×C₄	semidirect
28. C₂×C₆₇⋊C₁₆	C₂×C₄	semidirect
39. M₃₂×C₆₇	C₂×C₄	semidirect
41. C₂×C₁₆×C₆₇	C₂×C₄	semidirect
96. C₆₇	#33329	semidirect
101. #504	Dic₆₇	semidirect
124. M₃₂	C₂×Dic₆₇	semidirect
127. C₂×C₁₆	C₂×Dic₆₇	semidirect

Maximal subgroups

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

subgroup	index
#22573312	2
#22568635	2
#22561813	2
C₂×#16863173	2
C₆₇×#3407	2
#33329	67