Subgroups — #31009325

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	#22562117	C₂	1	4	yes	maximal, complemented
3	C₁₆×Dic₁₃₄	C₂	1	4	yes	maximal, complemented
4	#22561967	C₂	1	2	yes	maximal
5	#22561965	C₂	1	1	yes	maximal, characteristic
6	#22561805	C₂	1	2	yes	maximal
7	C₂×C₄×C₁₆×C₆₇	C₂	1	1	yes	maximal, characteristic
8	#16863953	C₄	1	1	yes	characteristic
9	C₂×#14523983	C₂²	1	2	yes	—
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	C₄⋊(C₆₇⋊C₁₆)	C₂²	1	4	yes	—
16	Dic₆₇⋊C₁₆	C₂²	1	8	yes	—
17	#16863171	C₂²	1	4	yes	—
18	C₁₆×Dic₆₇	C₂²	1	4	yes	—
19	C₂²×C₁₆×C₆₇	C₂²	1	1	yes	characteristic
20	C₂²×C₁₆×C₆₇	C₂²	1	1	yes	characteristic
21	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
22	C₄×C₁₆×C₆₇	C₂²	1	2	yes	—
23	C₄×C₁₆×C₆₇	C₂²	1	2	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	#14523983	—	2	4	no	—
31	C₂×C₆₇⋊C₁₆	C₂³	1	4	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	C₂×C₁₆×C₆₇	—	2	2	no	—
40	C₂×C₁₆×C₆₇	Q₈	1	4	yes	complemented
41	C₂×C₁₆×C₆₇	C₂³	1	2	yes	—
42	C₂×C₁₆×C₆₇	C₂³	1	1	yes	characteristic
43	C₂×C₁₆×C₆₇	C₂³	1	1	yes	characteristic
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	4	no	—
55	C₂²×C₄×C₆₇	C₂×C₈	1	2	yes	—
56	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
57	C₂×C₈×C₆₇	—	2	2	no	—
58	C₂×C₈×C₆₇	C₂²×C₄	1	2	yes	—
59	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
60	C₂×C₈×C₆₇	C₂×Q₈	1	2	yes	—
61	C₂×C₈×C₆₇	#41	1	2	yes	—
62	C₂×C₈×C₆₇	C₂⁴	1	1	yes	characteristic
63	C₄²×C₆₇	C₂×C₈	1	1	yes	characteristic
64	C₄²×C₆₇	C₂×C₈	1	1	yes	characteristic
65	C₄²×C₆₇	C₂²×C₄	1	2	yes	—
66	C₁₆×C₆₇	—	2	2	no	—
67	C₁₆×C₆₇	C₂×Q₈	1	2	yes	—
68	C₁₆×C₆₇	C₂×Q₈	1	2	yes	—
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	—	67	1	no	—
102	#2108	—	67	2	no	—
103	C₂×C₄×C₁₆	D₆₇	1	1	yes	characteristic
104	C₆₇	#33474	1	1	yes	characteristic, complemented, Sylow
105	#533	—	67	1	no	—
106	C₂×M₃₂	—	67	2	no	—
107	C₂²×C₁₆	C₂×D₆₇	1	1	yes	characteristic
108	C₂²×C₁₆	C₂×D₆₇	1	1	yes	characteristic
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	—	67	4	no	—
117	#346	—	67	4	no	—
118	C₄×C₁₆	—	67	4	no	—
119	C₄×C₁₆	C₂×D₆₇	1	2	yes	—
120	C₄×C₁₆	C₂×D₆₇	1	2	yes	—
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	—	134	4	no	—
131	C₂×C₁₆	—	2	2	no	—
132	C₂×C₁₆	—	67	4	no	—
133	C₂×C₁₆	Dic₁₃₄	1	4	yes	complemented
134	C₂×C₁₆	C₂²×D₆₇	1	2	yes	—
135	C₂×C₁₆	C₂²×D₆₇	1	1	yes	characteristic
136	C₂×C₁₆	C₂²×D₆₇	1	1	yes	characteristic
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	C₂×Dic₁₃₄	1	2	yes	—
152	C₂×C₈	#11761628	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	2	no	—
163	C₁₆	—	134	4	no	—
164	C₁₆	C₂×Dic₁₃₄	1	2	yes	—
165	C₁₆	C₂×Dic₁₃₄	1	2	yes	—
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	C₄×Dic₁₃₄	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	#14524049	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	C₄×Dic₁₃₄	1	2	yes	—
179	C₈	C₂²×Dic₁₃₄	1	1	yes	characteristic
180	C₈	C₂×#11761628	1	1	yes	characteristic
181	C₂²	—	2	2	no	—
182	C₂²	C₈×Dic₁₃₄	1	2	yes	—
183	C₂²	C₂²×C₈×D₆₇	1	1	yes	characteristic
184	C₄	—	2	2	no	—
185	C₄	—	134	8	no	—
186	C₄	C₈×Dic₁₃₄	1	2	yes	—
187	C₄	#16863812	1	4	yes	—
188	C₄	C₂×C₄×Dic₁₃₄	1	1	yes	characteristic
189	C₄	C₂×#14524049	1	1	yes	characteristic
190	C₂	—	2	2	no	—
191	C₂	C₂×C₈×Dic₁₃₄	1	1	yes	characteristic
192	C₂	C₂×#16863812	1	2	yes	—

	subgroup	G/H
center	C₂×C₁₆	C₂²×D₆₇
commutator, socle	C₂²×C₆₇	C₂³×C₈
Frattini	C₂×C₈	C₂³×D₆₇
Fitting	C₂×C₄×C₁₆×C₆₇	C₂
prime radical	C₂³×C₆₇	C₂²×C₈
cosocle	C₂×C₈×C₆₇	C₂⁴
hypercenter	C₂×C₄×C₁₆	D₆₇
hypocenter	C₆₇	#33474
LCS	G ▷ C₂²×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂×C₁₆ ◁ C₂×C₄×C₁₆

Complements

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

N	H ≅ G/N	type
2. #22562117	C₂	semidirect
3. C₁₆×Dic₁₃₄	C₂	semidirect
40. C₂×C₁₆×C₆₇	Q₈	semidirect
104. C₆₇	#33474	semidirect
133. C₂×C₁₆	Dic₁₃₄	semidirect

Maximal subgroups

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

subgroup	index
#22569143	2
#22562117	2
C₁₆×Dic₁₃₄	2
#22561967	2
#22561965	2
#22561805	2
C₂×C₄×C₁₆×C₆₇	2
#33474	67