Subgroups — #31009209

Order 17152.

646 conjugacy classes of subgroups.

The subgroup lattice has 208 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22569077	C₂	1	2	yes	maximal
2	#22562080	C₂	1	4	yes	maximal, complemented
3	Q₃₂×Dic₆₇	C₂	1	4	yes	maximal, complemented
4	#22561993	C₂	1	1	yes	maximal, characteristic
5	#22561929	C₂	1	2	yes	maximal
6	#22561805	C₂	1	1	yes	maximal, characteristic
7	C₂²×Q₃₂×C₆₇	C₂	1	1	yes	maximal, characteristic
8	#16863874	—	2	2	no	—
9	C₂×#14523983	C₂²	1	1	yes	characteristic
10	#16863503	—	2	4	no	—
11	Q₁₆×Dic₆₇	C₂²	1	8	yes	—
12	#16863434	C₂²	1	2	yes	—
13	#16863405	—	2	2	no	—
14	#16863351	C₂²	1	1	yes	characteristic
15	#16863224	C₂²	1	2	yes	—
16	#16863222	C₂²	1	2	yes	—
17	#16863200	C₂²	1	8	yes	—
18	#16863171	C₂²	1	2	yes	—
19	C₁₆×Dic₆₇	C₂²	1	2	yes	—
20	C₂²×Q₁₆×C₆₇	C₂²	1	2	yes	—
21	C₂×Q₃₂×C₆₇	C₄	1	8	yes	complemented
22	C₂×Q₃₂×C₆₇	C₂²	1	2	yes	—
23	C₂×Q₃₂×C₆₇	C₂²	1	2	yes	—
24	C₂²×C₁₆×C₆₇	C₂²	1	1	yes	characteristic
25	C₂×#11761612	D₄	1	1	yes	characteristic
26	Q₈×Dic₆₇	—	2	8	no	—
27	Q₈×Dic₆₇	—	4	4	no	—
28	#14524002	—	2	2	no	—
29	#14524002	—	4	4	no	—
30	#14524002	D₄	1	1	yes	characteristic
31	#14523983	—	2	2	no	—
32	C₂×C₆₇⋊C₁₆	C₂³	1	2	yes	—
33	#14523977	C₂³	1	4	yes	—
34	#14523975	—	2	2	no	—
35	#14523967	—	2	8	no	—
36	#14523953	D₄	1	2	yes	—
37	C₈×Dic₆₇	C₂³	1	2	yes	—
38	C₂²×Q₈×C₆₇	—	2	2	no	—
39	C₂×Q₁₆×C₆₇	—	2	2	no	—
40	C₂×Q₁₆×C₆₇	—	2	4	no	—
41	C₂×Q₁₆×C₆₇	C₂×C₄	1	8	yes	—
42	C₂×Q₁₆×C₆₇	C₂³	1	4	yes	—
43	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
44	Q₃₂×C₆₇	C₂×C₄	1	8	yes	—
45	Q₃₂×C₆₇	C₂×C₄	1	8	yes	—
46	C₂×C₁₆×C₆₇	C₂×C₄	1	4	yes	—
47	C₂×C₁₆×C₆₇	C₂³	1	1	yes	characteristic
48	C₂×C₁₆×C₆₇	C₂³	1	1	yes	characteristic
49	#11761612	—	2	2	no	—
50	C₂×C₆₇⋊C₈	C₂×D₄	1	2	yes	—
51	C₄⋊Dic₆₇	—	2	4	no	—
52	C₄⋊Dic₆₇	—	4	8	no	—
53	C₄⋊Dic₆₇	D₈	1	2	yes	—
54	C₂²⋊Dic₆₇	—	4	2	no	—
55	C₂²⋊Dic₆₇	D₈	1	2	yes	—
56	C₄×Dic₆₇	—	4	2	no	—
57	C₄×Dic₆₇	—	4	8	no	—
58	C₄×Dic₆₇	C₂×D₄	1	2	yes	—
59	C₆₇⋊C₁₆	—	2	2	no	—
60	C₂×Q₈×C₆₇	—	2	8	no	—
61	C₂×Q₈×C₆₇	—	2	4	no	—
62	C₂×Q₈×C₆₇	—	4	2	no	—
63	C₂×Q₈×C₆₇	—	4	4	no	—
64	C₂²×C₄×C₆₇	—	4	2	no	—
65	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
66	Q₁₆×C₆₇	—	2	8	no	—
67	Q₁₆×C₆₇	—	2	4	no	—
68	Q₁₆×C₆₇	C₂²×C₄	1	8	yes	—
69	C₂×C₈×C₆₇	C₂²×C₄	1	2	yes	—
70	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
71	C₂×C₈×C₆₇	#41	1	2	yes	—
72	C₂×C₈×C₆₇	C₂⁴	1	1	yes	characteristic
73	C₁₆×C₆₇	C₂²×C₄	1	2	yes	—
74	C₁₆×C₆₇	C₂²×C₄	1	2	yes	—
75	C₂³×C₆₇	C₂×D₈	1	1	yes	characteristic
76	Q₈×C₆₇	—	2	8	no	—
77	Q₈×C₆₇	—	4	4	no	—
78	Q₈×C₆₇	—	4	8	no	—
79	C₂×C₄×C₆₇	—	4	4	no	—
80	C₂×C₄×C₆₇	—	4	8	no	—
81	C₂×C₄×C₆₇	C₄×D₄	1	2	yes	—
82	C₂×C₄×C₆₇	C₂×D₈	1	1	yes	characteristic
83	C₂×C₄×C₆₇	#135	1	2	yes	—
84	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
85	C₂×Dic₆₇	—	4	4	no	—
86	C₂×Dic₆₇	C₂×D₈	1	4	yes	—
87	C₈×C₆₇	C₄×D₄	1	2	yes	—
88	C₈×C₆₇	C₂³×C₄	1	1	yes	characteristic
89	C₈×C₆₇	C₂×#41	1	1	yes	characteristic
90	C₆₇⋊C₈	—	2	2	no	—
91	C₂²×C₆₇	—	2	2	no	—
92	C₂²×C₆₇	C₄×D₈	1	2	yes	—
93	C₂²×C₆₇	C₂²×D₈	1	1	yes	characteristic
94	C₄×C₆₇	—	4	8	no	—
95	C₄×C₆₇	C₄×D₈	1	2	yes	—
96	C₄×C₆₇	C₂×C₄×D₄	1	1	yes	characteristic
97	C₄×C₆₇	C₂×#135	1	1	yes	characteristic
98	Dic₆₇	—	2	4	no	—
99	Dic₆₇	—	8	4	no	—
100	#33458	—	67	1	no	maximal, Sylow
101	C₂×C₆₇	—	2	2	no	—
102	C₂×C₆₇	C₂×C₄×D₈	1	1	yes	characteristic
103	C₂×C₆₇	#3417	1	2	yes	—
104	C₂²×Q₃₂	D₆₇	1	1	yes	characteristic
105	#2950	—	67	2	no	—
106	#2176	—	67	4	no	—
107	C₄×Q₃₂	—	67	4	no	—
108	#2158	—	67	1	no	—
109	#2141	—	67	2	no	—
110	#2108	—	67	1	no	—
111	C₆₇	#33458	1	1	yes	characteristic, complemented, Sylow
112	C₂²×Q₁₆	C₂×D₆₇	1	2	yes	—
113	#519	—	134	2	no	—
114	C₂×Q₃₂	Dic₆₇	1	8	yes	complemented
115	C₂×Q₃₂	C₂×D₆₇	1	2	yes	—
116	C₂×Q₃₂	C₂×D₆₇	1	2	yes	—
117	C₂×M₃₂	—	67	1	no	—
118	C₂²×C₁₆	C₂×D₆₇	1	1	yes	characteristic
119	#441	—	134	4	no	—
120	C₄×Q₁₆	—	67	8	no	—
121	#427	—	67	2	no	—
122	#419	—	134	2	no	—
123	#405	—	67	1	no	—
124	#367	—	67	2	no	—
125	#366	—	67	2	no	—
126	#358	—	67	8	no	—
127	#346	—	67	2	no	—
128	C₄×C₁₆	—	67	2	no	—
129	C₂²×Q₈	—	2	2	no	—
130	C₂×Q₁₆	—	2	2	no	—
131	C₂×Q₁₆	—	2	4	no	—
132	C₂×Q₁₆	C₂×Dic₆₇	1	8	yes	—
133	C₂×Q₁₆	C₂²×D₆₇	1	4	yes	—
134	C₂×M₁₆	—	67	1	no	—
135	C₂²×C₈	C₂²×D₆₇	1	1	yes	characteristic
136	C₄×Q₈	—	268	4	no	—
137	C₄×Q₈	—	134	8	no	—
138	#117	—	268	4	no	—
139	#117	—	67	1	no	—
140	#117	—	134	2	no	—
141	Q₃₂	C₂×Dic₆₇	1	8	yes	—
142	Q₃₂	C₂×Dic₆₇	1	8	yes	—
143	M₃₂	—	134	2	no	—
144	C₂×C₁₆	—	67	2	no	—
145	C₂×C₁₆	C₂×Dic₆₇	1	4	yes	—
146	C₂×C₁₆	C₂²×D₆₇	1	1	yes	characteristic
147	C₂×C₁₆	C₂²×D₆₇	1	1	yes	characteristic
148	#107	—	67	4	no	—
149	#106	—	134	2	no	—
150	#103	—	134	8	no	—
151	#97	—	67	2	no	—
152	C₄×C₈	—	67	2	no	—
153	C₂×Q₈	—	2	8	no	—
154	C₂×Q₈	—	2	4	no	—
155	C₂×Q₈	—	4	2	no	—
156	C₂×Q₈	—	4	4	no	—
157	C₂²×C₄	—	4	2	no	—
158	C₂²×C₄	D₄×D₆₇	1	1	yes	characteristic
159	Q₁₆	—	2	8	no	—
160	Q₁₆	—	2	4	no	—
161	Q₁₆	C₂²×Dic₆₇	1	8	yes	—
162	M₁₆	—	134	2	no	—
163	C₂×C₈	—	67	2	no	—
164	C₂×C₈	C₂²×Dic₆₇	1	2	yes	—
165	C₂×C₈	D₄×D₆₇	1	1	yes	characteristic
166	C₂×C₈	#11761627	1	2	yes	—
167	C₂×C₈	C₂³×D₆₇	1	1	yes	characteristic
168	C₄⋊C₄	—	268	8	no	—
169	C₄⋊C₄	—	67	2	no	—
170	C₄⋊C₄	—	134	4	no	—
171	C₂²⋊C₄	—	268	2	no	—
172	C₂²⋊C₄	—	67	2	no	—
173	C₄²	—	268	8	no	—
174	C₄²	—	268	2	no	—
175	C₄²	—	67	2	no	—
176	C₁₆	—	134	2	no	—
177	C₁₆	C₂²×Dic₆₇	1	2	yes	—
178	C₁₆	C₂²×Dic₆₇	1	2	yes	—
179	C₂³	D₈×D₆₇	1	1	yes	characteristic
180	Q₈	—	2	8	no	—
181	Q₈	—	4	4	no	—
182	Q₈	—	4	8	no	—
183	C₂×C₄	—	4	4	no	—
184	C₂×C₄	—	4	8	no	—
185	C₂×C₄	—	268	4	no	—
186	C₂×C₄	—	67	4	no	—
187	C₂×C₄	D₄×Dic₆₇	1	2	yes	—
188	C₂×C₄	D₈×D₆₇	1	1	yes	characteristic
189	C₂×C₄	#14524062	1	2	yes	—
190	C₂×C₄	C₂×D₄×D₆₇	1	1	yes	characteristic
191	C₈	—	134	2	no	—
192	C₈	D₄×Dic₆₇	1	2	yes	—
193	C₈	C₂³×Dic₆₇	1	1	yes	characteristic
194	C₈	C₂×#11761627	1	1	yes	characteristic
195	C₂²	—	2	2	no	—
196	C₂²	D₈×Dic₆₇	1	2	yes	—
197	C₂²	C₂×D₈×D₆₇	1	1	yes	characteristic
198	C₄	—	4	8	no	—
199	C₄	—	536	4	no	—
200	C₄	—	134	4	no	—
201	C₄	D₈×Dic₆₇	1	2	yes	—
202	C₄	C₂×D₄×Dic₆₇	1	1	yes	characteristic
203	C₄	C₂×#14524062	1	1	yes	characteristic
204	C₂	—	2	2	no	—
205	C₂	C₂×D₈×Dic₆₇	1	1	yes	characteristic
206	C₂	#22573358	1	2	yes	—

	subgroup	G/H
center	C₂²	C₂×D₈×D₆₇
commutator	C₈×C₆₇	C₂³×C₄
Frattini	C₂×C₈	C₂³×D₆₇
Fitting	C₂²×Q₃₂×C₆₇	C₂
socle	C₂²×C₆₇	C₂²×D₈
prime radical	C₂³×C₆₇	C₂×D₈
cosocle	C₂×C₈×C₆₇	C₂⁴
hypercenter	C₂²×Q₃₂	D₆₇
hypocenter	C₆₇	#33458
LCS	G ▷ C₈×C₆₇ ▷ C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂²×Q₃₂

Complements

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

N	H ≅ G/N	type
2. #22562080	C₂	semidirect
3. Q₃₂×Dic₆₇	C₂	semidirect
21. C₂×Q₃₂×C₆₇	C₄	semidirect
111. C₆₇	#33458	semidirect
114. C₂×Q₃₂	Dic₆₇	semidirect

Maximal subgroups

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

subgroup	index
#22569077	2
#22562080	2
Q₃₂×Dic₆₇	2
#22561993	2
#22561929	2
#22561805	2
C₂²×Q₃₂×C₆₇	2
#33458	67