Subgroups — #31009210

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	C₂²×Dic₅₃₆	C₂	1	1	yes	maximal, characteristic
2	#22569078	C₂	1	2	yes	maximal
3	#22562081	C₂	1	4	yes	maximal, complemented
4	C₄×Dic₅₃₆	C₂	1	4	yes	maximal, complemented
5	#22561993	C₂	1	1	yes	maximal, characteristic
6	#22561930	C₂	1	2	yes	maximal
7	C₆₇×#2108	C₂	1	1	yes	maximal, characteristic
8	C₂²×Dic₂₆₈	C₂²	1	2	yes	—
9	#16863873	—	2	2	no	—
10	C₂×Dic₅₃₆	C₄	1	8	yes	complemented
11	C₂×Dic₅₃₆	C₂²	1	2	yes	—
12	C₂×Dic₅₃₆	C₂²	1	2	yes	—
13	#16863504	—	2	4	no	—
14	C₄×Dic₂₆₈	C₂²	1	8	yes	—
15	#16863434	C₂²	1	2	yes	—
16	#16863406	—	2	2	no	—
17	#16863224	C₂²	1	2	yes	—
18	#16863222	C₂²	1	2	yes	—
19	#16863201	C₂²	1	8	yes	—
20	C₂×M₃₂×C₆₇	C₂²	1	1	yes	characteristic
21	C₂²×C₁₆×C₆₇	C₂²	1	1	yes	characteristic
22	#405×C₆₇	C₂²	1	1	yes	characteristic
23	#346×C₆₇	C₂²	1	2	yes	—
24	C₄×C₁₆×C₆₇	C₂²	1	2	yes	—
25	C₂²×Dic₁₃₄	—	2	2	no	—
26	C₂×Dic₂₆₈	—	2	2	no	—
27	C₂×Dic₂₆₈	—	2	4	no	—
28	C₂×Dic₂₆₈	C₂×C₄	1	8	yes	—
29	C₂×Dic₂₆₈	C₂³	1	4	yes	—
30	C₄×Dic₁₃₄	—	2	8	no	—
31	#14524008	—	4	4	no	—
32	#14524002	—	2	2	no	—
33	#14523999	—	4	4	no	—
34	Dic₅₃₆	C₂×C₄	1	8	yes	—
35	Dic₅₃₆	C₂×C₄	1	8	yes	—
36	#14523977	C₂³	1	4	yes	—
37	#14523975	—	2	2	no	—
38	#14523968	—	2	8	no	—
39	C₂×M₁₆×C₆₇	D₄	1	1	yes	characteristic
40	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
41	#117×C₆₇	D₄	1	1	yes	characteristic
42	M₃₂×C₆₇	—	2	2	no	—
43	C₂×C₁₆×C₆₇	C₂×C₄	1	4	yes	—
44	C₂×C₁₆×C₆₇	C₂³	1	2	yes	—
45	C₂×C₁₆×C₆₇	C₂³	1	1	yes	characteristic
46	C₂×C₁₆×C₆₇	C₂³	1	1	yes	characteristic
47	#97×C₆₇	D₄	1	2	yes	—
48	C₄×C₈×C₆₇	C₂³	1	2	yes	—
49	C₂×Dic₁₃₄	—	2	8	no	—
50	C₂×Dic₁₃₄	—	2	4	no	—
51	C₂×Dic₁₃₄	—	4	2	no	—
52	C₂×Dic₁₃₄	—	4	4	no	—
53	C₂²×Dic₆₇	—	4	2	no	—
54	Dic₂₆₈	—	2	8	no	—
55	Dic₂₆₈	—	2	4	no	—
56	Dic₂₆₈	C₂²×C₄	1	8	yes	—
57	C₄⋊Dic₆₇	—	2	4	no	—
58	Dic₆₇⋊C₄	—	4	8	no	—
59	C₂²⋊Dic₆₇	—	4	2	no	—
60	C₄×Dic₆₇	—	4	8	no	—
61	C₄×Dic₆₇	—	4	2	no	—
62	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
63	M₁₆×C₆₇	—	2	2	no	—
64	C₂×C₈×C₆₇	C₂²×C₄	1	2	yes	—
65	C₂×C₈×C₆₇	C₂×D₄	1	2	yes	—
66	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
67	C₂×C₈×C₆₇	#41	1	2	yes	—
68	C₂×C₈×C₆₇	C₂⁴	1	1	yes	characteristic
69	C₄⋊C₄×C₆₇	D₈	1	2	yes	—
70	C₂²⋊C₄×C₆₇	D₈	1	2	yes	—
71	C₄²×C₆₇	C₂×D₄	1	2	yes	—
72	C₁₆×C₆₇	—	2	2	no	—
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	C₂×C₄×C₆₇	C₄×D₄	1	2	yes	—
77	C₂×C₄×C₆₇	C₂×D₈	1	4	yes	—
78	C₂×C₄×C₆₇	C₂×D₈	1	1	yes	characteristic
79	C₂×C₄×C₆₇	#135	1	2	yes	—
80	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
81	C₂×Dic₆₇	—	4	4	no	—
82	C₂×Dic₆₇	—	4	8	no	—
83	C₂×Dic₆₇	—	4	4	no	—
84	Dic₁₃₄	—	2	8	no	—
85	Dic₁₃₄	—	4	4	no	—
86	Dic₁₃₄	—	4	8	no	—
87	C₈×C₆₇	—	2	2	no	—
88	C₈×C₆₇	C₄×D₄	1	2	yes	—
89	C₈×C₆₇	C₂³×C₄	1	1	yes	characteristic
90	C₈×C₆₇	C₂×#41	1	1	yes	characteristic
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	—	2	4	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	—	4	8	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	—	67	1	no	—
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	D₆₇	1	1	yes	characteristic
111	C₆₇	#33458	1	1	yes	characteristic, complemented, Sylow
112	C₂²×Q₁₆	—	67	2	no	—
113	#519	—	134	2	no	—
114	C₂×Q₃₂	—	67	8	no	—
115	C₂×Q₃₂	—	67	2	no	—
116	C₂×Q₃₂	—	67	2	no	—
117	C₂×M₃₂	C₂×D₆₇	1	1	yes	characteristic
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	C₂×D₆₇	1	1	yes	characteristic
124	#367	—	67	2	no	—
125	#366	—	67	2	no	—
126	#358	—	67	8	no	—
127	#346	C₂×D₆₇	1	2	yes	—
128	C₄×C₁₆	C₂×D₆₇	1	2	yes	—
129	C₂²×Q₈	—	134	2	no	—
130	C₂×Q₁₆	—	67	8	no	—
131	C₂×Q₁₆	—	67	4	no	—
132	C₂×Q₁₆	—	134	2	no	—
133	C₂×Q₁₆	—	134	4	no	—
134	C₂×M₁₆	D₂₆₈	1	1	yes	characteristic
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	—	134	2	no	—
140	#117	D₂₆₈	1	1	yes	characteristic
141	Q₃₂	—	67	8	no	—
142	Q₃₂	—	67	8	no	—
143	M₃₂	—	2	2	no	—
144	C₂×C₁₆	C₄×D₆₇	1	4	yes	—
145	C₂×C₁₆	C₂²×D₆₇	1	2	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	D₂₆₈	1	2	yes	—
152	C₄×C₈	C₂²×D₆₇	1	2	yes	—
153	C₂×Q₈	—	268	2	no	—
154	C₂×Q₈	—	268	4	no	—
155	C₂×Q₈	—	134	8	no	—
156	C₂×Q₈	—	134	4	no	—
157	C₂²×C₄	—	268	2	no	—
158	C₂²×C₄	C₂×D₂₆₈	1	1	yes	characteristic
159	Q₁₆	—	67	8	no	—
160	Q₁₆	—	134	8	no	—
161	Q₁₆	—	134	4	no	—
162	M₁₆	—	2	2	no	—
163	C₂×C₈	C₂×C₄×D₆₇	1	2	yes	—
164	C₂×C₈	C₂×D₂₆₈	1	2	yes	—
165	C₂×C₈	C₂×D₂₆₈	1	1	yes	characteristic
166	C₂×C₈	#11761628	1	2	yes	—
167	C₂×C₈	C₂³×D₆₇	1	1	yes	characteristic
168	C₄⋊C₄	—	268	8	no	—
169	C₄⋊C₄	—	134	4	no	—
170	C₄⋊C₄	D₅₃₆	1	2	yes	—
171	C₂²⋊C₄	—	268	2	no	—
172	C₂²⋊C₄	D₅₃₆	1	2	yes	—
173	C₄²	—	268	8	no	—
174	C₄²	—	268	2	no	—
175	C₄²	C₂×D₂₆₈	1	2	yes	—
176	C₁₆	—	2	2	no	—
177	C₁₆	C₂×C₄×D₆₇	1	2	yes	—
178	C₁₆	C₂×C₄×D₆₇	1	2	yes	—
179	C₂³	C₂×D₅₃₆	1	1	yes	characteristic
180	Q₈	—	268	4	no	—
181	Q₈	—	268	8	no	—
182	Q₈	—	134	8	no	—
183	C₂×C₄	—	268	4	no	—
184	C₂×C₄	—	268	8	no	—
185	C₂×C₄	—	268	4	no	—
186	C₂×C₄	C₄×D₂₆₈	1	2	yes	—
187	C₂×C₄	C₂×D₅₃₆	1	4	yes	—
188	C₂×C₄	C₂×D₅₃₆	1	1	yes	characteristic
189	C₂×C₄	#14524065	1	2	yes	—
190	C₂×C₄	C₂²×D₂₆₈	1	1	yes	characteristic
191	C₈	—	2	2	no	—
192	C₈	C₄×D₂₆₈	1	2	yes	—
193	C₈	C₂²×C₄×D₆₇	1	1	yes	characteristic
194	C₈	C₂×#11761628	1	1	yes	characteristic
195	C₂²	—	2	2	no	—
196	C₂²	C₄×D₅₃₆	1	2	yes	—
197	C₂²	C₂²×D₅₃₆	1	1	yes	characteristic
198	C₄	—	2	4	no	—
199	C₄	—	268	8	no	—
200	C₄	—	536	4	no	—
201	C₄	C₄×D₅₃₆	1	2	yes	—
202	C₄	C₂×C₄×D₂₆₈	1	1	yes	characteristic
203	C₄	C₂×#14524065	1	1	yes	characteristic
204	C₂	—	2	2	no	—
205	C₂	C₂×C₄×D₅₃₆	1	1	yes	characteristic
206	C₂	#22573362	1	2	yes	—

	subgroup	G/H
center	C₂²	C₂²×D₅₃₆
commutator	C₈×C₆₇	C₂³×C₄
Frattini	C₂×C₈	C₂³×D₆₇
Fitting	C₆₇×#2108	C₂
socle	C₂²×C₆₇	C₂²×D₈
prime radical	C₂³×C₆₇	C₂×D₈
cosocle	C₂×C₈×C₆₇	C₂⁴
hypercenter	#2108	D₆₇
hypocenter	C₆₇	#33458
LCS	G ▷ C₈×C₆₇ ▷ C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂² ◁ #117 ◁ #405 ◁ #2108

Complements

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

N	H ≅ G/N	type
3. #22562081	C₂	semidirect
4. C₄×Dic₅₃₆	C₂	semidirect
10. C₂×Dic₅₃₆	C₄	semidirect
111. C₆₇	#33458	semidirect

Maximal subgroups

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

subgroup	index
C₂²×Dic₅₃₆	2
#22569078	2
#22562081	2
C₄×Dic₅₃₆	2
#22561993	2
#22561930	2
C₆₇×#2108	2
#33458	67