Subgroups — #31009321

Order 17152.

506 conjugacy classes of subgroups.

The subgroup lattice has 194 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	#22562114	C₂	1	4	yes	maximal, complemented
3	#22562111	C₂	1	4	yes	maximal, complemented
4	#22561969	C₂	1	2	yes	maximal
5	#22561808	C₂	1	2	yes	maximal
6	C₂×C₄×C₆₇⋊C₁₆	C₂	1	1	yes	maximal, characteristic
7	C₆₇×#2150	C₂	1	1	yes	maximal, characteristic
8	#16863953	C₄	1	1	yes	characteristic
9	C₂²×C₆₇⋊C₁₆	C₂²	1	1	yes	characteristic
10	C₂²×C₆₇⋊C₁₆	C₂²	1	1	yes	characteristic
11	#16863526	C₄	1	4	yes	—
12	C₈×Dic₁₃₄	C₂²	1	4	yes	—
13	#16863422	C₄	1	2	yes	—
14	#16863420	C₄	1	1	yes	characteristic
15	#16863351	C₂²	1	2	yes	—
16	Dic₆₇⋊C₁₆	C₂²	1	8	yes	—
17	#16863171	C₂²	1	4	yes	—
18	C₄×C₆₇⋊C₁₆	C₂²	1	2	yes	—
19	C₄×C₆₇⋊C₁₆	C₂²	1	2	yes	—
20	C₁₆×Dic₆₇	C₂²	1	4	yes	—
21	C₂×M₃₂×C₆₇	C₂²	1	2	yes	—
22	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
23	C₄⋊C₁₆×C₆₇	C₂²	1	4	yes	—
24	C₂×#11761612	C₂×C₄	1	2	yes	—
25	#14524043	C₈	1	2	yes	—
26	#14524035	C₈	1	2	yes	—
27	#14524022	C₈	1	4	yes	—
28	C₄×Dic₁₃₄	C₂×C₄	1	4	yes	—
29	#14524002	C₂×C₄	1	2	yes	—
30	C₂×C₆₇⋊C₁₆	—	2	2	no	—
31	C₂×C₆₇⋊C₁₆	Q₈	1	4	yes	complemented
32	C₂×C₆₇⋊C₁₆	C₂³	1	2	yes	—
33	C₂×C₆₇⋊C₁₆	C₂³	1	1	yes	characteristic
34	C₂×C₆₇⋊C₁₆	C₂³	1	1	yes	characteristic
35	C₄⋊(C₆₇⋊C₈)	C₂×C₄	1	4	yes	—
36	Dic₆₇⋊C₈	C₂×C₄	1	8	yes	—
37	#14523953	C₂×C₄	1	4	yes	—
38	C₈×Dic₆₇	C₂³	1	4	yes	—
39	C₂²×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
40	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
41	C₂×C₄²×C₆₇	C₂×C₄	1	1	yes	characteristic
42	M₃₂×C₆₇	—	2	4	no	—
43	C₂×C₁₆×C₆₇	C₂³	1	4	yes	—
44	C₄×C₈×C₆₇	C₂×C₄	1	2	yes	—
45	C₄×C₈×C₆₇	C₂³	1	2	yes	—
46	C₂×Dic₁₃₄	C₂×C₈	1	4	yes	—
47	#11761612	—	2	4	no	—
48	C₂×C₆₇⋊C₈	C₂²×C₄	1	4	yes	—
49	C₄⋊Dic₆₇	C₂×C₈	1	4	yes	—
50	C₄⋊Dic₆₇	C₂×C₈	1	4	yes	—
51	Dic₆₇⋊C₄	C₂×C₈	1	8	yes	—
52	C₂²⋊Dic₆₇	C₂×C₈	1	4	yes	—
53	C₄×Dic₆₇	C₂²×C₄	1	4	yes	—
54	C₆₇⋊C₁₆	—	2	2	no	—
55	C₆₇⋊C₁₆	C₂×Q₈	1	2	yes	—
56	C₆₇⋊C₁₆	C₂×Q₈	1	2	yes	—
57	C₂²×C₄×C₆₇	C₂×C₈	1	2	yes	—
58	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
59	C₂×C₈×C₆₇	—	2	2	no	—
60	C₂×C₈×C₆₇	C₂²×C₄	1	2	yes	—
61	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
62	C₂×C₈×C₆₇	C₂×Q₈	1	2	yes	—
63	C₂×C₈×C₆₇	#41	1	2	yes	—
64	C₂×C₈×C₆₇	C₂⁴	1	1	yes	characteristic
65	C₄²×C₆₇	C₂×C₈	1	1	yes	characteristic
66	C₄²×C₆₇	C₂×C₈	1	1	yes	characteristic
67	C₄²×C₆₇	C₂²×C₄	1	2	yes	—
68	C₁₆×C₆₇	—	2	4	no	—
69	C₂³×C₆₇	C₂²×C₈	1	1	yes	characteristic
70	C₂×C₄×C₆₇	—	2	4	no	—
71	C₂×C₄×C₆₇	C₄×Q₈	1	2	yes	—
72	C₂×C₄×C₆₇	C₂²×C₈	1	2	yes	—
73	C₂×C₄×C₆₇	C₂²×C₈	1	1	yes	characteristic
74	C₂×C₄×C₆₇	C₂²×C₈	1	2	yes	—
75	C₂×C₄×C₆₇	#131	1	2	yes	—
76	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
77	C₂×Dic₆₇	C₂²×C₈	1	8	yes	—
78	Dic₁₃₄	—	2	8	no	—
79	C₈×C₆₇	—	2	2	no	—
80	C₈×C₆₇	C₄×Q₈	1	2	yes	—
81	C₈×C₆₇	C₂²×Q₈	1	1	yes	characteristic
82	C₈×C₆₇	C₂×#41	1	1	yes	characteristic
83	C₆₇⋊C₈	—	2	4	no	—
84	C₂²×C₆₇	—	2	2	no	—
85	C₂²×C₆₇	C₈×Q₈	1	2	yes	—
86	C₂²×C₆₇	C₂³×C₈	1	1	yes	characteristic
87	C₄×C₆₇	—	2	2	no	—
88	C₄×C₆₇	C₈×Q₈	1	2	yes	—
89	C₄×C₆₇	#504	1	4	yes	—
90	C₄×C₆₇	C₂×C₄×Q₈	1	1	yes	characteristic
91	C₄×C₆₇	C₂×#131	1	1	yes	characteristic
92	Dic₆₇	—	2	8	no	—
93	#33474	—	67	1	no	maximal, Sylow
94	C₂×C₆₇	—	2	2	no	—
95	C₂×C₆₇	C₂×C₈×Q₈	1	1	yes	characteristic
96	C₂×C₆₇	C₂×#504	1	2	yes	—
97	#2960	—	67	1	no	—
98	#2183	—	67	4	no	—
99	Q₈×C₁₆	—	67	4	no	—
100	#2151	—	67	2	no	—
101	#2150	D₆₇	1	1	yes	characteristic
102	#2108	—	67	2	no	—
103	C₂×C₄×C₁₆	—	67	1	no	—
104	C₆₇	#33474	1	1	yes	characteristic, complemented, Sylow
105	#533	—	67	1	no	—
106	C₂×M₃₂	C₂×D₆₇	1	2	yes	—
107	C₂²×C₁₆	—	67	1	no	—
108	C₂²×C₁₆	—	67	1	no	—
109	#446	—	67	4	no	—
110	C₈×Q₈	—	67	4	no	—
111	#424	—	67	2	no	—
112	#423	—	67	1	no	—
113	#405	—	67	2	no	—
114	C₂×C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
115	C₄⋊C₁₆	—	67	8	no	—
116	C₄⋊C₁₆	C₂×D₆₇	1	4	yes	—
117	#346	—	67	4	no	—
118	C₄×C₁₆	—	67	4	no	—
119	C₄×C₁₆	—	67	2	no	—
120	C₄×C₁₆	—	67	2	no	—
121	C₂×M₁₆	—	67	2	no	—
122	C₂²×C₈	C₄×D₆₇	1	1	yes	characteristic
123	C₂²×C₈	C₂²×D₆₇	1	1	yes	characteristic
124	C₄⋊Q₈	—	67	2	no	—
125	#125	—	67	2	no	—
126	C₂²⋊Q₈	—	67	4	no	—
127	C₄×Q₈	—	67	4	no	—
128	#117	—	67	2	no	—
129	C₂×C₄²	C₄×D₆₇	1	1	yes	characteristic
130	M₃₂	—	2	4	no	—
131	C₂×C₁₆	—	67	4	no	—
132	C₂×C₁₆	—	67	1	no	—
133	C₂×C₁₆	—	67	1	no	—
134	C₂×C₁₆	—	67	2	no	—
135	C₂×C₁₆	—	134	2	no	—
136	C₂×C₁₆	C₂²×D₆₇	1	4	yes	—
137	C₄⋊C₈	—	67	8	no	—
138	C₄⋊C₈	—	67	4	no	—
139	#97	—	67	4	no	—
140	C₄×C₈	—	67	4	no	—
141	C₄×C₈	C₄×D₆₇	1	2	yes	—
142	C₄×C₈	C₂²×D₆₇	1	2	yes	—
143	C₂×Q₈	—	67	4	no	—
144	C₂²×C₄	C₈×D₆₇	1	2	yes	—
145	C₂²×C₄	C₂×C₄×D₆₇	1	1	yes	characteristic
146	M₁₆	—	134	4	no	—
147	C₂×C₈	—	2	2	no	—
148	C₂×C₈	—	67	4	no	—
149	C₂×C₈	C₂×C₄×D₆₇	1	2	yes	—
150	C₂×C₈	C₂×C₄×D₆₇	1	1	yes	characteristic
151	C₂×C₈	Q₈×D₆₇	1	2	yes	—
152	C₂×C₈	#11761627	1	2	yes	—
153	C₂×C₈	C₂³×D₆₇	1	1	yes	characteristic
154	C₄⋊C₄	—	67	4	no	—
155	C₄⋊C₄	—	67	8	no	—
156	C₄⋊C₄	—	67	4	no	—
157	C₂²⋊C₄	—	67	4	no	—
158	C₄²	—	67	4	no	—
159	C₄²	C₈×D₆₇	1	1	yes	characteristic
160	C₄²	C₈×D₆₇	1	1	yes	characteristic
161	C₄²	C₂×C₄×D₆₇	1	2	yes	—
162	C₁₆	—	2	4	no	—
163	C₁₆	—	67	2	no	—
164	C₁₆	—	67	2	no	—
165	C₁₆	—	134	2	no	—
166	C₂³	C₂×C₈×D₆₇	1	1	yes	characteristic
167	Q₈	—	134	8	no	—
168	C₂×C₄	—	2	4	no	—
169	C₂×C₄	—	67	8	no	—
170	C₂×C₄	#14524008	1	2	yes	—
171	C₂×C₄	C₂×C₈×D₆₇	1	2	yes	—
172	C₂×C₄	C₂×C₈×D₆₇	1	1	yes	characteristic
173	C₂×C₄	C₂×C₈×D₆₇	1	2	yes	—
174	C₂×C₄	#14524051	1	2	yes	—
175	C₂×C₄	C₂²×C₄×D₆₇	1	1	yes	characteristic
176	C₈	—	2	2	no	—
177	C₈	—	134	4	no	—
178	C₈	#14524008	1	2	yes	—
179	C₈	C₂×Q₈×D₆₇	1	1	yes	characteristic
180	C₈	C₂×#11761627	1	1	yes	characteristic
181	C₂²	—	2	2	no	—
182	C₂²	#16863520	1	2	yes	—
183	C₂²	C₂²×C₈×D₆₇	1	1	yes	characteristic
184	C₄	—	2	2	no	—
185	C₄	—	134	8	no	—
186	C₄	#16863520	1	2	yes	—
187	C₄	#16863814	1	4	yes	—
188	C₄	C₂×#14524008	1	1	yes	characteristic
189	C₄	C₂×#14524051	1	1	yes	characteristic
190	C₂	—	2	2	no	—
191	C₂	C₂×#16863520	1	1	yes	characteristic
192	C₂	C₂×#16863814	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	#33474
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. #22562114	C₂	semidirect
3. #22562111	C₂	semidirect
31. C₂×C₆₇⋊C₁₆	Q₈	semidirect
104. C₆₇	#33474	semidirect

Maximal subgroups

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

subgroup	index
#22569143	2
#22562114	2
#22562111	2
#22561969	2
#22561808	2
C₂×C₄×C₆₇⋊C₁₆	2
C₆₇×#2150	2
#33474	67