Subgroups — #30771227

Order 17152.

310 conjugacy classes of subgroups.

The subgroup lattice has 180 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16863226	C₂	1	1	yes	maximal, characteristic
2	C₂×#16863171	C₂	1	1	yes	maximal, characteristic
3	#22560090	C₂	1	2	yes	maximal
4	#22558304	C₂	1	2	yes	maximal
5	C₆₇×#2158	C₂	1	1	yes	maximal, characteristic
6	C₂²×C₆₇⋊C₁₆	C₂²	1	1	yes	characteristic
7	C₂×#14523979	C₂²	1	2	yes	—
8	#16863352	—	2	2	no	—
9	C₂×C₈×Dic₆₇	C₂²	1	1	yes	characteristic
10	#16863226	C₄	1	4	yes	complemented
11	#16863171	C₄	1	4	yes	complemented
12	#16863155	—	2	2	no	—
13	C₂²×C₁₆×C₆₇	C₂²	1	1	yes	characteristic
14	#427×C₆₇	C₂²	1	2	yes	—
15	#367×C₆₇	C₄	1	2	yes	complemented
16	#366×C₆₇	C₄	1	2	yes	complemented
17	C₂×#11761612	—	2	2	no	—
18	C₂²×C₆₇⋊C₈	D₄	1	1	yes	characteristic
19	C₂×C₄×Dic₆₇	D₄	1	1	yes	characteristic
20	C₂×C₆₇⋊C₁₆	—	2	1	no	—
21	C₂×C₆₇⋊C₁₆	—	2	1	no	—
22	C₂×C₆₇⋊C₁₆	C₂×C₄	1	2	yes	—
23	#14523979	—	2	2	no	—
24	#14523979	C₂×C₄	1	4	yes	—
25	#14523954	—	4	2	no	—
26	C₄×C₆₇⋊C₈	—	4	2	no	—
27	C₈×Dic₆₇	C₂×C₄	1	2	yes	—
28	C₈×Dic₆₇	Q₈	1	2	yes	—
29	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
30	#117×C₆₇	—	2	2	no	—
31	C₂×C₁₆×C₆₇	C₂×C₄	1	2	yes	—
32	C₂×C₁₆×C₆₇	C₂×C₄	1	2	yes	—
33	C₂×C₁₆×C₆₇	C₂×C₄	1	1	yes	characteristic
34	C₂×C₁₆×C₆₇	C₂×C₄	1	1	yes	characteristic
35	#107×C₆₇	C₂×C₄	1	4	yes	—
36	#106×C₆₇	—	2	2	no	—
37	C₂²×Dic₆₇	—	2	1	no	—
38	#11761612	—	2	4	no	—
39	#11761612	—	4	2	no	—
40	C₂×C₆₇⋊C₈	—	4	2	no	—
41	C₂×C₆₇⋊C₈	C₄⋊C₄	1	2	yes	—
42	C₂×C₆₇⋊C₈	D₈	1	2	yes	—
43	C₂×C₆₇⋊C₈	Q₁₆	1	2	yes	—
44	C₄×Dic₆₇	—	2	1	no	—
45	C₄×Dic₆₇	C₄⋊C₄	1	2	yes	—
46	C₆₇⋊C₁₆	—	2	2	no	—
47	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
48	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
49	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
50	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
51	C₂×C₈×C₆₇	C₂×Q₈	1	1	yes	characteristic
52	C₂×C₈×C₆₇	#41	1	1	yes	characteristic
53	C₂×C₈×C₆₇	#41	1	1	yes	characteristic
54	C₄⋊C₄×C₆₇	—	2	4	no	—
55	C₂²⋊C₄×C₆₇	—	4	2	no	—
56	C₄²×C₆₇	—	4	2	no	—
57	C₁₆×C₆₇	C₄²	1	2	yes	—
58	C₁₆×C₆₇	C₄²	1	2	yes	—
59	C₂³×C₆₇	#135	1	1	yes	characteristic
60	C₂×C₄×C₆₇	—	4	4	no	—
61	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
62	C₂×C₄×C₆₇	C₄×D₄	1	1	yes	characteristic
63	C₂×C₄×C₆₇	C₄×D₄	1	1	yes	characteristic
64	C₂×C₄×C₆₇	C₂×D₈	1	1	yes	characteristic
65	C₂×C₄×C₆₇	C₂×Q₁₆	1	1	yes	characteristic
66	C₂×C₄×C₆₇	#135	1	1	yes	characteristic
67	C₂×Dic₆₇	—	2	2	no	—
68	C₂×Dic₆₇	—	4	2	no	—
69	C₈×C₆₇	C₂×C₄²	1	1	yes	characteristic
70	C₈×C₆₇	#117	1	1	yes	characteristic
71	C₈×C₆₇	C₄×Q₈	1	1	yes	characteristic
72	C₈×C₆₇	C₄×Q₈	1	1	yes	characteristic
73	C₆₇⋊C₈	—	4	4	no	—
74	C₆₇⋊C₈	#107	1	4	yes	—
75	C₂²×C₆₇	—	2	2	no	—
76	C₂²×C₆₇	#427	1	1	yes	characteristic
77	C₂²×C₆₇	C₄×D₈	1	1	yes	characteristic
78	C₂²×C₆₇	C₄×Q₁₆	1	1	yes	characteristic
79	C₄×C₆₇	—	8	4	no	—
80	C₄×C₆₇	C₄×C₄⋊C₄	1	1	yes	characteristic
81	C₄×C₆₇	C₂×#107	1	1	yes	characteristic
82	C₄×C₆₇	C₄×D₈	1	1	yes	characteristic
83	C₄×C₆₇	C₄×Q₁₆	1	1	yes	characteristic
84	Dic₆₇	—	4	2	no	—
85	#12671	—	67	1	no	maximal, Sylow
86	C₂×C₆₇	—	2	1	no	—
87	C₂×C₆₇	—	2	1	no	—
88	C₂×C₆₇	C₄×#107	1	1	yes	characteristic
89	C₂×C₆₇	#2179	1	2	yes	—
90	C₂×#368	—	67	1	no	—
91	#2158	D₆₇	1	1	yes	characteristic
92	C₂×#346	—	67	1	no	—
93	#1772	—	67	2	no	—
94	#1382	—	67	2	no	—
95	C₆₇	#12671	1	1	yes	characteristic, complemented, Sylow
96	C₂²×C₁₆	—	67	1	no	—
97	C₂²×C₁₆	C₂×D₆₇	1	1	yes	characteristic
98	C₂×#108	—	67	2	no	—
99	#427	C₂×D₆₇	1	2	yes	—
100	#405	—	134	2	no	—
101	C₂×C₄×C₈	—	67	1	no	—
102	#368	—	67	4	no	—
103	#367	Dic₆₇	1	2	yes	complemented
104	#366	Dic₆₇	1	2	yes	complemented
105	#346	—	67	4	no	—
106	#339	—	134	2	no	—
107	C₂×M₁₆	—	134	2	no	—
108	C₂²×C₈	—	67	1	no	—
109	C₂²×C₈	C₂²×D₆₇	1	1	yes	characteristic
110	#117	—	2	2	no	—
111	C₂×C₄²	—	67	1	no	—
112	C₂×C₁₆	—	67	2	no	—
113	C₂×C₁₆	—	134	1	no	—
114	C₂×C₁₆	—	134	1	no	—
115	C₂×C₁₆	C₄×D₆₇	1	2	yes	—
116	C₂×C₁₆	C₄×D₆₇	1	2	yes	—
117	C₂×C₁₆	C₂×Dic₆₇	1	1	yes	characteristic
118	C₂×C₁₆	C₂×Dic₆₇	1	1	yes	characteristic
119	#108	—	67	4	no	—
120	#108	—	134	2	no	—
121	#107	C₂×Dic₆₇	1	4	yes	—
122	#106	—	2	2	no	—
123	#97	—	268	2	no	—
124	C₄×C₈	—	268	2	no	—
125	C₄×C₈	—	67	2	no	—
126	C₄×C₈	—	67	2	no	—
127	C₂²×C₄	—	134	1	no	—
128	C₂²×C₄	D₄×D₆₇	1	1	yes	characteristic
129	M₁₆	—	268	2	no	—
130	M₁₆	—	134	4	no	—
131	C₂×C₈	—	268	2	no	—
132	C₂×C₈	—	67	2	no	—
133	C₂×C₈	—	67	2	no	—
134	C₂×C₈	—	67	2	no	—
135	C₂×C₈	C₂×C₄×D₆₇	1	1	yes	characteristic
136	C₂×C₈	C₂×C₄×D₆₇	1	1	yes	characteristic
137	C₂×C₈	C₂²×Dic₆₇	1	1	yes	characteristic
138	C₂×C₈	Q₈×D₆₇	1	1	yes	characteristic
139	C₂×C₈	#11761627	1	1	yes	characteristic
140	C₂×C₈	#11761629	1	1	yes	characteristic
141	C₄⋊C₄	—	2	4	no	—
142	C₂²⋊C₄	—	4	2	no	—
143	C₄²	—	4	2	no	—
144	C₄²	—	67	2	no	—
145	C₄²	—	134	1	no	—
146	C₁₆	—	134	2	no	—
147	C₁₆	C₄×Dic₆₇	1	2	yes	—
148	C₁₆	C₄×Dic₆₇	1	2	yes	—
149	C₂³	#14524062	1	1	yes	characteristic
150	C₂×C₄	—	4	4	no	—
151	C₂×C₄	—	268	2	no	—
152	C₂×C₄	—	134	2	no	—
153	C₂×C₄	C₄⋊C₄×D₆₇	1	1	yes	characteristic
154	C₂×C₄	#14524005	1	1	yes	characteristic
155	C₂×C₄	D₄×Dic₆₇	1	1	yes	characteristic
156	C₂×C₄	D₈×D₆₇	1	1	yes	characteristic
157	C₂×C₄	Q₁₆×D₆₇	1	1	yes	characteristic
158	C₂×C₄	#14524066	1	1	yes	characteristic
159	C₈	—	268	4	no	—
160	C₈	—	67	4	no	—
161	C₈	C₂×C₄×Dic₆₇	1	1	yes	characteristic
162	C₈	#14524001	1	1	yes	characteristic
163	C₈	#14524008	1	1	yes	characteristic
164	C₈	Q₈×Dic₆₇	1	1	yes	characteristic
165	C₂²	—	2	2	no	—
166	C₂²	#16863433	1	1	yes	characteristic
167	C₂²	D₈×Dic₆₇	1	1	yes	characteristic
168	C₂²	#16863490	1	1	yes	characteristic
169	C₄	—	8	4	no	—
170	C₄	—	268	2	no	—
171	C₄	C₄⋊C₄×Dic₆₇	1	1	yes	characteristic
172	C₄	#107×D₆₇	1	1	yes	characteristic
173	C₄	#16863478	1	1	yes	characteristic
174	C₄	Q₁₆×Dic₆₇	1	1	yes	characteristic
175	C₂	—	2	1	no	—
176	C₂	—	2	1	no	—
177	C₂	#107×Dic₆₇	1	1	yes	characteristic
178	C₂	#22562095	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	D₈×D₆₇
commutator	C₈×C₆₇	C₂×C₄²
Frattini	C₂²×C₈	C₂²×D₆₇
Fitting	C₆₇×#2158	C₂
socle	C₂²×C₆₇	C₄×D₈
prime radical	C₂³×C₆₇	#135
cosocle	C₂²×C₈×C₆₇	C₂³
hypercenter	#2158	D₆₇
hypocenter	C₆₇	#12671
LCS	G ▷ C₈×C₆₇ ▷ C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ #2158

Complements

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

N	H ≅ G/N	type
10. #16863226	C₄	semidirect
11. #16863171	C₄	semidirect
15. #367×C₆₇	C₄	semidirect
16. #366×C₆₇	C₄	semidirect
95. C₆₇	#12671	semidirect
103. #367	Dic₆₇	semidirect
104. #366	Dic₆₇	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16863226	2
C₂×#16863171	2
#22560090	2
#22558304	2
C₆₇×#2158	2
#12671	67