Subgroups — #30772188

Order 17152.

314 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	C₂×#16863217	C₂	1	1	yes	maximal, characteristic
2	C₄×#14523953	C₂	1	1	yes	maximal, characteristic
3	#22558292	C₂	1	2	yes	maximal
4	#22558288	C₂	1	2	yes	maximal
5	C₂×#364×C₆₇	C₂	1	1	yes	maximal, characteristic
6	C₂×#14523983	C₂²	1	2	yes	—
7	C₂×#14523953	C₄	1	1	yes	characteristic
8	C₂×#14523953	C₄	1	1	yes	characteristic
9	C₂×#14523953	C₂²	1	1	yes	characteristic
10	C₂×#14523953	C₂²	1	1	yes	characteristic
11	C₂×C₄×C₆₇⋊C₈	C₄	1	1	yes	characteristic
12	C₄²×Dic₆₇	C₄	1	1	yes	characteristic
13	#16863217	—	2	2	no	—
14	C₂×M₃₂×C₆₇	C₂²	1	2	yes	—
15	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
16	#364×C₆₇	—	2	2	no	—
17	C₂²×C₆₇⋊C₈	C₂×C₄	1	1	yes	characteristic
18	C₂²×C₆₇⋊C₈	C₂×C₄	1	1	yes	characteristic
19	C₂×C₄×Dic₆₇	—	2	2	no	—
20	C₂×C₄×Dic₆₇	C₂×C₄	1	1	yes	characteristic
21	C₂×C₄×Dic₆₇	C₂×C₄	1	1	yes	characteristic
22	#14523983	—	2	2	no	—
23	#14523983	—	2	2	no	—
24	C₂×C₆₇⋊C₁₆	—	2	2	no	—
25	#14523953	—	2	2	no	—
26	#14523953	—	2	1	no	—
27	#14523953	—	2	1	no	—
28	#14523953	—	2	1	no	—
29	#14523953	—	2	1	no	—
30	#14523953	C₈	1	4	yes	—
31	C₄×C₆₇⋊C₈	—	2	2	no	—
32	C₂²×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
33	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
34	C₂×C₄²×C₆₇	C₂×C₄	1	1	yes	characteristic
35	M₃₂×C₆₇	—	2	2	no	—
36	M₃₂×C₆₇	—	2	2	no	—
37	C₂×C₁₆×C₆₇	—	2	2	no	—
38	C₄×C₈×C₆₇	D₄	1	2	yes	—
39	C₄×C₈×C₆₇	Q₈	1	2	yes	—
40	C₂²×Dic₆₇	—	2	1	no	—
41	C₂²×Dic₆₇	M₁₆	1	2	yes	—
42	C₂×C₆₇⋊C₈	—	2	1	no	—
43	C₂×C₆₇⋊C₈	—	2	1	no	—
44	C₂×C₆₇⋊C₈	—	2	1	no	—
45	C₂×C₆₇⋊C₈	—	2	1	no	—
46	C₂×C₆₇⋊C₈	C₂×C₈	1	2	yes	—
47	C₂×C₆₇⋊C₈	M₁₆	1	2	yes	—
48	C₄×Dic₆₇	—	2	1	no	—
49	C₄×Dic₆₇	—	2	1	no	—
50	C₄×Dic₆₇	—	2	1	no	—
51	C₄×Dic₆₇	—	4	4	no	—
52	C₄×Dic₆₇	C₂×C₈	1	2	yes	—
53	C₆₇⋊C₁₆	—	4	2	no	—
54	C₂²×C₄×C₆₇	—	2	1	no	—
55	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
56	C₂×C₈×C₆₇	C₄⋊C₄	1	2	yes	—
57	C₂×C₈×C₆₇	C₂×C₈	1	2	yes	—
58	C₂×C₈×C₆₇	M₁₆	1	2	yes	—
59	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
60	C₂×C₈×C₆₇	C₂×Q₈	1	1	yes	characteristic
61	C₂×C₈×C₆₇	#41	1	1	yes	characteristic
62	C₂×C₈×C₆₇	#41	1	1	yes	characteristic
63	C₂×C₈×C₆₇	#41	1	1	yes	characteristic
64	C₂×C₈×C₆₇	#41	1	1	yes	characteristic
65	C₄²×C₆₇	—	2	1	no	—
66	C₄²×C₆₇	C₄⋊C₄	1	2	yes	—
67	C₁₆×C₆₇	—	4	2	no	—
68	C₂³×C₆₇	C₂×M₁₆	1	1	yes	characteristic
69	C₂×C₄×C₆₇	—	2	4	no	—
70	C₂×C₄×C₆₇	—	2	2	no	—
71	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
72	C₂×C₄×C₆₇	#117	1	1	yes	characteristic
73	C₂×C₄×C₆₇	#117	1	1	yes	characteristic
74	C₂×C₄×C₆₇	C₂²×C₈	1	1	yes	characteristic
75	C₂×C₄×C₆₇	C₂×M₁₆	1	1	yes	characteristic
76	C₂×C₄×C₆₇	C₂×M₁₆	1	1	yes	characteristic
77	C₂×Dic₆₇	—	2	2	no	—
78	C₂×Dic₆₇	—	2	2	no	—
79	C₂×Dic₆₇	—	4	2	no	—
80	C₂×Dic₆₇	—	4	2	no	—
81	C₈×C₆₇	C₄⋊C₈	1	4	yes	—
82	C₈×C₆₇	#124	1	1	yes	characteristic
83	C₈×C₆₇	#124	1	1	yes	characteristic
84	C₈×C₆₇	#125	1	1	yes	characteristic
85	C₈×C₆₇	#125	1	1	yes	characteristic
86	C₆₇⋊C₈	—	2	2	no	—
87	C₆₇⋊C₈	—	2	2	no	—
88	C₂²×C₆₇	—	2	2	no	—
89	C₂²×C₆₇	#423	1	1	yes	characteristic
90	C₂²×C₆₇	#431	1	1	yes	characteristic
91	C₂²×C₆₇	#432	1	1	yes	characteristic
92	C₄×C₆₇	—	2	4	no	—
93	C₄×C₆₇	#382	1	1	yes	characteristic
94	C₄×C₆₇	C₂×C₄⋊C₈	1	1	yes	characteristic
95	C₄×C₆₇	#431	1	1	yes	characteristic
96	C₄×C₆₇	#432	1	1	yes	characteristic
97	Dic₆₇	—	4	2	no	—
98	Dic₆₇	—	4	2	no	—
99	#12882	—	67	1	no	maximal, Sylow
100	C₂×C₆₇	—	2	1	no	—
101	C₂×C₆₇	—	2	1	no	—
102	C₂×C₆₇	#1831	1	1	yes	characteristic
103	C₂×C₆₇	#2165	1	2	yes	—
104	C₂×#364	—	67	1	no	—
105	C₂×#364	D₆₇	1	1	yes	characteristic
106	C₄×#97	—	67	1	no	—
107	#1377	—	67	2	no	—
108	#1375	—	67	2	no	—
109	C₆₇	#12882	1	1	yes	characteristic, complemented, Sylow
110	C₂×M₃₂	—	67	2	no	—
111	C₂×M₃₂	C₂×D₆₇	1	2	yes	—
112	C₂×#97	—	67	1	no	—
113	C₂×#97	—	67	1	no	—
114	C₂×#97	—	67	1	no	—
115	C₂×#97	—	67	1	no	—
116	C₂×C₄×C₈	—	67	1	no	—
117	C₂×C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
118	C₄³	—	67	1	no	—
119	#364	—	2	2	no	—
120	#364	—	134	2	no	—
121	C₂²×C₈	—	67	1	no	—
122	C₂²×C₈	—	67	1	no	—
123	C₂²×C₈	C₄×D₆₇	1	1	yes	characteristic
124	C₂²×C₈	C₂²×D₆₇	1	1	yes	characteristic
125	C₂×C₄²	—	67	1	no	—
126	C₂×C₄²	—	67	1	no	—
127	C₂×C₄²	—	134	2	no	—
128	C₂×C₄²	C₄×D₆₇	1	1	yes	characteristic
129	M₃₂	—	2	2	no	—
130	M₃₂	—	2	2	no	—
131	M₃₂	—	134	2	no	—
132	M₃₂	—	134	2	no	—
133	C₂×C₁₆	—	2	2	no	—
134	C₂×C₁₆	—	134	2	no	—
135	#97	—	67	4	no	—
136	#97	—	134	2	no	—
137	#97	—	134	1	no	—
138	#97	—	134	1	no	—
139	#97	—	134	1	no	—
140	#97	—	134	1	no	—
141	C₄×C₈	—	134	2	no	—
142	C₄×C₈	Dic₁₃₄	1	2	yes	—
143	C₄×C₈	C₂²⋊D₆₇	1	2	yes	—
144	C₂²×C₄	—	2	1	no	—
145	C₂²×C₄	—	67	2	no	—
146	C₂²×C₄	—	134	1	no	—
147	C₂²×C₄	C₂×C₄×D₆₇	1	1	yes	characteristic
148	C₂×C₈	—	67	2	no	—
149	C₂×C₈	—	67	2	no	—
150	C₂×C₈	—	134	1	no	—
151	C₂×C₈	—	134	1	no	—
152	C₂×C₈	—	134	1	no	—
153	C₂×C₈	—	134	1	no	—
154	C₂×C₈	Dic₆₇⋊C₄	1	2	yes	—
155	C₂×C₈	C₈×D₆₇	1	2	yes	—
156	C₂×C₈	#11761611	1	2	yes	—
157	C₂×C₈	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
158	C₂×C₈	C₂×Dic₁₃₄	1	1	yes	characteristic
159	C₂×C₈	#11761627	1	1	yes	characteristic
160	C₂×C₈	#11761627	1	1	yes	characteristic
161	C₂×C₈	#11761629	1	1	yes	characteristic
162	C₂×C₈	#11761629	1	1	yes	characteristic
163	C₄²	—	2	1	no	—
164	C₄²	—	268	4	no	—
165	C₄²	—	67	2	no	—
166	C₄²	—	134	1	no	—
167	C₄²	—	134	1	no	—
168	C₄²	—	134	1	no	—
169	C₄²	Dic₆₇⋊C₄	1	2	yes	—
170	C₁₆	—	4	2	no	—
171	C₁₆	—	268	2	no	—
172	C₂³	M₁₆×D₆₇	1	1	yes	characteristic
173	C₂×C₄	—	2	4	no	—
174	C₂×C₄	—	2	2	no	—
175	C₂×C₄	—	268	2	no	—
176	C₂×C₄	—	268	2	no	—
177	C₂×C₄	—	134	2	no	—
178	C₂×C₄	—	134	2	no	—
179	C₂×C₄	C₂×Dic₆₇⋊C₄	1	1	yes	characteristic
180	C₂×C₄	#14524001	1	1	yes	characteristic
181	C₂×C₄	#14524001	1	1	yes	characteristic
182	C₂×C₄	C₂×C₈×D₆₇	1	1	yes	characteristic
183	C₂×C₄	M₁₆×D₆₇	1	1	yes	characteristic
184	C₂×C₄	C₂×#11761611	1	1	yes	characteristic
185	C₈	—	134	2	no	—
186	C₈	—	134	2	no	—
187	C₈	Dic₆₇⋊C₈	1	4	yes	—
188	C₈	#14524030	1	1	yes	characteristic
189	C₈	#14524032	1	1	yes	characteristic
190	C₈	#14524034	1	1	yes	characteristic
191	C₈	#14524034	1	1	yes	characteristic
192	C₂²	—	2	2	no	—
193	C₂²	#16863419	1	1	yes	characteristic
194	C₂²	#16863448	1	1	yes	characteristic
195	C₂²	#16863452	1	1	yes	characteristic
196	C₄	—	2	4	no	—
197	C₄	—	268	2	no	—
198	C₄	—	268	2	no	—
199	C₄	#16863261	1	1	yes	characteristic
200	C₄	C₂×Dic₆₇⋊C₈	1	1	yes	characteristic
201	C₄	#16863448	1	1	yes	characteristic
202	C₄	#16863452	1	1	yes	characteristic
203	C₂	—	2	1	no	—
204	C₂	—	2	1	no	—
205	C₂	#22560325	1	1	yes	characteristic
206	C₂	#22562020	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	#14524001
commutator	C₂×C₄×C₆₇	C₂²×C₈
Frattini	C₂²×C₈	C₂²×D₆₇
Fitting	C₂×#364×C₆₇	C₂
socle	C₂²×C₆₇	#432
prime radical	C₂³×C₆₇	C₂×M₁₆
cosocle	C₂²×C₈×C₆₇	C₂³
hypercenter	C₂×#364	D₆₇
hypocenter	C₆₇	#12882
LCS	G ▷ C₂×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₈ ◁ C₂×#364

Complements

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

N	H ≅ G/N	type
109. C₆₇	#12882	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16863217	2
C₄×#14523953	2
#22558292	2
#22558288	2
C₂×#364×C₆₇	2
#12882	67