Subgroups — #31009328

Order 17152.

490 conjugacy classes of subgroups.

The subgroup lattice has 186 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22569143	C₂	1	1	yes	maximal, characteristic
2	#22562116	C₂	1	8	yes	maximal, complemented
3	#22561969	C₂	1	2	yes	maximal
4	#22561808	C₂	1	2	yes	maximal
5	C₂×#16863172	C₂	1	1	yes	maximal, characteristic
6	C₆₇×#2150	C₂	1	1	yes	maximal, characteristic
7	#16863953	C₄	1	1	yes	characteristic
8	C₂²×C₆₇⋊C₁₆	C₂²	1	1	yes	characteristic
9	C₂²×C₆₇⋊C₁₆	C₂²	1	1	yes	characteristic
10	#16863526	C₄	1	4	yes	—
11	C₈×Dic₁₃₄	C₂²	1	4	yes	—
12	#16863422	C₄	1	2	yes	—
13	#16863420	C₄	1	1	yes	characteristic
14	#16863351	C₂²	1	2	yes	—
15	Dic₆₇⋊C₁₆	C₂²	1	8	yes	—
16	#16863172	C₂²	1	4	yes	—
17	#16863171	C₂²	1	4	yes	—
18	C₁₆×Dic₆₇	C₂²	1	4	yes	—
19	C₂×M₃₂×C₆₇	C₂²	1	2	yes	—
20	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
21	C₄⋊C₁₆×C₆₇	C₂²	1	4	yes	—
22	C₂×#11761612	C₂×C₄	1	2	yes	—
23	#14524043	C₈	1	2	yes	—
24	#14524035	C₈	1	2	yes	—
25	#14524022	C₈	1	4	yes	—
26	C₄×Dic₁₃₄	C₂×C₄	1	4	yes	—
27	#14524002	C₂×C₄	1	2	yes	—
28	C₂×C₆₇⋊C₁₆	—	2	2	no	—
29	C₂×C₆₇⋊C₁₆	Q₈	1	4	yes	complemented
30	C₂×C₆₇⋊C₁₆	C₂³	1	2	yes	—
31	C₂×C₆₇⋊C₁₆	C₂³	1	2	yes	—
32	C₄⋊(C₆₇⋊C₈)	C₂×C₄	1	4	yes	—
33	Dic₆₇⋊C₈	C₂×C₄	1	8	yes	—
34	#14523953	C₂×C₄	1	4	yes	—
35	C₈×Dic₆₇	C₂³	1	4	yes	—
36	C₂²×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
37	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
38	C₂×C₄²×C₆₇	C₂×C₄	1	1	yes	characteristic
39	M₃₂×C₆₇	—	2	4	no	—
40	C₂×C₁₆×C₆₇	C₂³	1	4	yes	—
41	C₄×C₈×C₆₇	C₂×C₄	1	2	yes	—
42	C₄×C₈×C₆₇	C₂³	1	2	yes	—
43	C₂×Dic₁₃₄	C₂×C₈	1	4	yes	—
44	#11761612	—	2	4	no	—
45	C₂×C₆₇⋊C₈	C₂²×C₄	1	4	yes	—
46	C₄⋊Dic₆₇	C₂×C₈	1	4	yes	—
47	C₄⋊Dic₆₇	C₂×C₈	1	4	yes	—
48	Dic₆₇⋊C₄	C₂×C₈	1	8	yes	—
49	C₂²⋊Dic₆₇	C₂×C₈	1	4	yes	—
50	C₄×Dic₆₇	C₂²×C₄	1	4	yes	—
51	C₆₇⋊C₁₆	—	2	2	no	—
52	C₆₇⋊C₁₆	C₂×Q₈	1	4	yes	—
53	C₂²×C₄×C₆₇	C₂×C₈	1	2	yes	—
54	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
55	C₂×C₈×C₆₇	—	2	2	no	—
56	C₂×C₈×C₆₇	C₂²×C₄	1	2	yes	—
57	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
58	C₂×C₈×C₆₇	C₂×Q₈	1	2	yes	—
59	C₂×C₈×C₆₇	#41	1	2	yes	—
60	C₂×C₈×C₆₇	C₂⁴	1	1	yes	characteristic
61	C₄²×C₆₇	C₂×C₈	1	1	yes	characteristic
62	C₄²×C₆₇	C₂×C₈	1	1	yes	characteristic
63	C₄²×C₆₇	C₂²×C₄	1	2	yes	—
64	C₁₆×C₆₇	—	2	4	no	—
65	C₂³×C₆₇	C₂²×C₈	1	1	yes	characteristic
66	C₂×C₄×C₆₇	—	4	2	no	—
67	C₂×C₄×C₆₇	C₄×Q₈	1	2	yes	—
68	C₂×C₄×C₆₇	C₂²×C₈	1	2	yes	—
69	C₂×C₄×C₆₇	C₂²×C₈	1	2	yes	—
70	C₂×C₄×C₆₇	C₂²×C₈	1	1	yes	characteristic
71	C₂×C₄×C₆₇	#131	1	2	yes	—
72	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
73	C₂×Dic₆₇	C₂²×C₈	1	8	yes	—
74	Dic₁₃₄	—	4	4	no	—
75	C₈×C₆₇	—	2	2	no	—
76	C₈×C₆₇	C₄×Q₈	1	2	yes	—
77	C₈×C₆₇	C₂²×Q₈	1	1	yes	characteristic
78	C₈×C₆₇	C₂×#41	1	1	yes	characteristic
79	C₆₇⋊C₈	—	2	4	no	—
80	C₂²×C₆₇	—	2	2	no	—
81	C₂²×C₆₇	C₈×Q₈	1	2	yes	—
82	C₂²×C₆₇	C₂³×C₈	1	1	yes	characteristic
83	C₄×C₆₇	—	2	2	no	—
84	C₄×C₆₇	—	2	2	no	—
85	C₄×C₆₇	C₈×Q₈	1	2	yes	—
86	C₄×C₆₇	C₂×C₄×Q₈	1	1	yes	characteristic
87	C₄×C₆₇	C₂×#131	1	1	yes	characteristic
88	Dic₆₇	—	2	8	no	—
89	#33475	—	67	1	no	maximal, Sylow
90	C₂×C₆₇	—	2	2	no	—
91	C₂×C₆₇	C₂×C₈×Q₈	1	1	yes	characteristic
92	C₂×C₆₇	#3407	1	2	yes	—
93	#2960	—	67	1	no	—
94	#2183	—	67	8	no	—
95	#2151	—	67	2	no	—
96	#2150	D₆₇	1	1	yes	characteristic
97	#2108	—	67	2	no	—
98	C₂×#346	—	67	1	no	—
99	C₆₇	#33475	1	1	yes	characteristic, complemented, Sylow
100	#533	—	67	1	no	—
101	C₂×M₃₂	C₂×D₆₇	1	2	yes	—
102	C₂²×C₁₆	—	67	1	no	—
103	C₂²×C₁₆	—	67	1	no	—
104	#446	—	67	4	no	—
105	C₈×Q₈	—	67	4	no	—
106	#424	—	67	2	no	—
107	#423	—	67	1	no	—
108	#405	—	67	2	no	—
109	C₂×C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
110	C₄⋊C₁₆	—	67	8	no	—
111	C₄⋊C₁₆	C₂×D₆₇	1	4	yes	—
112	#346	—	67	4	no	—
113	#346	—	67	4	no	—
114	C₄×C₁₆	—	67	4	no	—
115	C₂×M₁₆	—	67	2	no	—
116	C₂²×C₈	C₄×D₆₇	1	1	yes	characteristic
117	C₂²×C₈	C₂²×D₆₇	1	1	yes	characteristic
118	C₄⋊Q₈	—	67	2	no	—
119	#125	—	67	2	no	—
120	C₂²⋊Q₈	—	67	4	no	—
121	C₄×Q₈	—	67	4	no	—
122	#117	—	67	2	no	—
123	C₂×C₄²	C₄×D₆₇	1	1	yes	characteristic
124	M₃₂	—	2	4	no	—
125	C₂×C₁₆	—	67	4	no	—
126	C₂×C₁₆	—	67	2	no	—
127	C₂×C₁₆	—	67	2	no	—
128	C₂×C₁₆	—	134	2	no	—
129	C₂×C₁₆	C₂²×D₆₇	1	4	yes	—
130	C₄⋊C₈	—	67	8	no	—
131	C₄⋊C₈	—	67	4	no	—
132	#97	—	67	4	no	—
133	C₄×C₈	—	67	4	no	—
134	C₄×C₈	C₄×D₆₇	1	2	yes	—
135	C₄×C₈	C₂²×D₆₇	1	2	yes	—
136	C₂×Q₈	—	67	4	no	—
137	C₂²×C₄	C₈×D₆₇	1	2	yes	—
138	C₂²×C₄	C₂×C₄×D₆₇	1	1	yes	characteristic
139	M₁₆	—	134	4	no	—
140	C₂×C₈	—	2	2	no	—
141	C₂×C₈	—	67	4	no	—
142	C₂×C₈	C₂×C₄×D₆₇	1	2	yes	—
143	C₂×C₈	C₂×C₄×D₆₇	1	1	yes	characteristic
144	C₂×C₈	Q₈×D₆₇	1	2	yes	—
145	C₂×C₈	#11761627	1	2	yes	—
146	C₂×C₈	C₂³×D₆₇	1	1	yes	characteristic
147	C₄⋊C₄	—	67	4	no	—
148	C₄⋊C₄	—	67	8	no	—
149	C₄⋊C₄	—	67	4	no	—
150	C₂²⋊C₄	—	67	4	no	—
151	C₄²	—	67	4	no	—
152	C₄²	C₈×D₆₇	1	1	yes	characteristic
153	C₄²	C₈×D₆₇	1	1	yes	characteristic
154	C₄²	C₂×C₄×D₆₇	1	2	yes	—
155	C₁₆	—	2	4	no	—
156	C₁₆	—	67	4	no	—
157	C₁₆	—	134	2	no	—
158	C₂³	C₂×C₈×D₆₇	1	1	yes	characteristic
159	Q₈	—	268	4	no	—
160	C₂×C₄	—	4	2	no	—
161	C₂×C₄	—	67	8	no	—
162	C₂×C₄	#14524008	1	2	yes	—
163	C₂×C₄	C₂×C₈×D₆₇	1	2	yes	—
164	C₂×C₄	C₂×C₈×D₆₇	1	2	yes	—
165	C₂×C₄	C₂×C₈×D₆₇	1	1	yes	characteristic
166	C₂×C₄	#14524051	1	2	yes	—
167	C₂×C₄	C₂²×C₄×D₆₇	1	1	yes	characteristic
168	C₈	—	2	2	no	—
169	C₈	—	134	4	no	—
170	C₈	#14524008	1	2	yes	—
171	C₈	C₂×Q₈×D₆₇	1	1	yes	characteristic
172	C₈	C₂×#11761627	1	1	yes	characteristic
173	C₂²	—	2	2	no	—
174	C₂²	#16863520	1	2	yes	—
175	C₂²	C₂²×C₈×D₆₇	1	1	yes	characteristic
176	C₄	—	2	2	no	—
177	C₄	—	2	2	no	—
178	C₄	—	134	8	no	—
179	C₄	#16863520	1	2	yes	—
180	C₄	C₂×#14524008	1	1	yes	characteristic
181	C₄	C₂×#14524051	1	1	yes	characteristic
182	C₂	—	2	2	no	—
183	C₂	C₂×#16863520	1	1	yes	characteristic
184	C₂	#22573311	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂×C₈	C₂³×D₆₇
commutator, socle	C₂²×C₆₇	C₂³×C₈
Fitting	C₆₇×#2150	C₂
prime radical	C₂³×C₆₇	C₂²×C₈
cosocle	C₂×C₈×C₆₇	C₂⁴
hypercenter	#2150	D₆₇
hypocenter	C₆₇	#33475
LCS	G ▷ C₂²×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂×C₈ ◁ #2150

Complements

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

N	H ≅ G/N	type
2. #22562116	C₂	semidirect
29. C₂×C₆₇⋊C₁₆	Q₈	semidirect
99. C₆₇	#33475	semidirect

Maximal subgroups

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

subgroup	index
#22569143	2
#22562116	2
#22561969	2
#22561808	2
C₂×#16863172	2
C₆₇×#2150	2
#33475	67