Subgroups — #30767199

Order 17152.

292 conjugacy classes of subgroups.

The subgroup lattice has 204 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	C₂×#16863217	C₂	1	1	yes	maximal, characteristic
3	#22558270	C₂	1	4	yes	maximal, complemented
4	C₆₇×#3160	C₂	1	1	yes	maximal, characteristic
5	C₂×#14523983	—	2	1	no	—
6	C₂×#14523983	—	2	1	no	—
7	#16863217	—	2	1	no	—
8	#16863217	—	2	1	no	—
9	#16863217	C₂²	1	2	yes	—
10	#16863217	C₂²	1	2	yes	—
11	#533×C₆₇	C₄	1	2	yes	—
12	#500×C₆₇	—	2	1	no	—
13	#499×C₆₇	C₄	1	2	yes	—
14	#499×C₆₇	C₂²	1	2	yes	—
15	#498×C₆₇	—	2	1	no	—
16	#427×C₆₇	—	2	1	no	—
17	#427×C₆₇	—	2	1	no	—
18	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
19	#14523983	—	2	2	no	—
20	#14523983	—	2	2	no	—
21	#14523983	—	4	1	no	—
22	#14523983	—	4	1	no	—
23	C₂×C₆₇⋊C₁₆	—	4	1	no	—
24	C₂×C₆₇⋊C₁₆	—	4	1	no	—
25	C₂²×C₈×C₆₇	D₄	1	1	yes	characteristic
26	C₂²×C₈×C₆₇	D₄	1	1	yes	characteristic
27	C₄⋊Q₈×C₆₇	—	2	2	no	—
28	#125×C₆₇	C₂×C₄	1	4	yes	—
29	C₂²⋊Q₈×C₆₇	—	4	2	no	—
30	C₄×Q₈×C₆₇	—	4	2	no	—
31	#117×C₆₇	—	2	2	no	—
32	C₂×C₄²×C₆₇	C₂×C₄	1	1	yes	characteristic
33	#107×C₆₇	—	2	2	no	—
34	#107×C₆₇	—	2	2	no	—
35	#106×C₆₇	—	2	2	no	—
36	#106×C₆₇	—	2	2	no	—
37	C₄×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
38	C₄×C₈×C₆₇	D₄	1	1	yes	characteristic
39	C₄×C₈×C₆₇	D₄	1	1	yes	characteristic
40	C₄×C₈×C₆₇	C₂³	1	1	yes	characteristic
41	C₆₇⋊C₁₆	—	4	2	no	—
42	C₆₇⋊C₁₆	—	4	2	no	—
43	C₂×Q₈×C₆₇	—	4	2	no	—
44	C₂²×C₄×C₆₇	—	2	1	no	—
45	C₂²×C₄×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
46	C₂×C₈×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
47	C₂×C₈×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
48	C₂×C₈×C₆₇	SD₁₆	1	1	yes	characteristic
49	C₂×C₈×C₆₇	SD₁₆	1	1	yes	characteristic
50	C₂×C₈×C₆₇	SD₁₆	1	1	yes	characteristic
51	C₂×C₈×C₆₇	SD₁₆	1	1	yes	characteristic
52	C₂×C₈×C₆₇	Q₁₆	1	1	yes	characteristic
53	C₂×C₈×C₆₇	Q₁₆	1	1	yes	characteristic
54	C₂×C₈×C₆₇	Q₁₆	1	1	yes	characteristic
55	C₂×C₈×C₆₇	Q₁₆	1	1	yes	characteristic
56	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
57	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
58	C₄⋊C₄×C₆₇	—	2	4	no	—
59	C₄⋊C₄×C₆₇	—	4	4	no	—
60	C₄⋊C₄×C₆₇	—	4	2	no	—
61	C₂²⋊C₄×C₆₇	—	4	2	no	—
62	C₄²×C₆₇	—	2	1	no	—
63	C₄²×C₆₇	—	4	2	no	—
64	C₄²×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
65	C₄²×C₆₇	C₂²×C₄	1	1	yes	characteristic
66	C₂³×C₆₇	#100	1	1	yes	characteristic
67	Q₈×C₆₇	—	8	4	no	—
68	C₂×C₄×C₆₇	—	2	1	no	—
69	C₂×C₄×C₆₇	—	2	1	no	—
70	C₂×C₄×C₆₇	—	4	2	no	—
71	C₂×C₄×C₆₇	—	4	4	no	—
72	C₂×C₄×C₆₇	#100	1	1	yes	characteristic
73	C₂×C₄×C₆₇	#103	1	1	yes	characteristic
74	C₂×C₄×C₆₇	#103	1	1	yes	characteristic
75	C₂×C₄×C₆₇	#103	1	1	yes	characteristic
76	C₂×C₄×C₆₇	#103	1	1	yes	characteristic
77	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
78	C₈×C₆₇	#103	1	1	yes	characteristic
79	C₈×C₆₇	#103	1	1	yes	characteristic
80	C₈×C₆₇	#103	1	1	yes	characteristic
81	C₈×C₆₇	#103	1	1	yes	characteristic
82	C₈×C₆₇	C₂×SD₁₆	1	1	yes	characteristic
83	C₈×C₆₇	C₂×SD₁₆	1	1	yes	characteristic
84	C₈×C₆₇	C₂×Q₁₆	1	1	yes	characteristic
85	C₈×C₆₇	C₂×Q₁₆	1	1	yes	characteristic
86	C₂²×C₆₇	—	4	1	no	—
87	C₂²×C₆₇	#333	1	1	yes	characteristic
88	C₂²×C₆₇	#333	1	1	yes	characteristic
89	C₂²×C₆₇	C₂×#100	1	1	yes	characteristic
90	C₄×C₆₇	—	2	2	no	—
91	C₄×C₆₇	—	4	1	no	—
92	C₄×C₆₇	—	8	4	no	—
93	C₄×C₆₇	#333	1	1	yes	characteristic
94	C₄×C₆₇	#333	1	1	yes	characteristic
95	C₄×C₆₇	C₂×#103	1	1	yes	characteristic
96	C₄×C₆₇	C₂×#103	1	1	yes	characteristic
97	#11956	—	67	1	no	maximal, Sylow
98	C₂×C₆₇	—	2	1	no	—
99	C₂×C₆₇	—	2	1	no	—
100	C₂×C₆₇	—	2	1	no	—
101	C₂×C₆₇	C₂×#333	1	1	yes	characteristic
102	#3160	D₆₇	1	1	yes	characteristic
103	C₂×#364	—	67	1	no	—
104	C₂×#364	—	67	1	no	—
105	#1366	—	67	4	no	—
106	C₆₇	#11956	1	1	yes	characteristic, complemented, Sylow
107	#533	Dic₆₇	1	2	yes	—
108	C₂×M₃₂	—	134	1	no	—
109	C₂×M₃₂	—	134	1	no	—
110	#500	—	2	1	no	—
111	#499	Dic₆₇	1	2	yes	—
112	#499	C₂×D₆₇	1	2	yes	—
113	#498	—	2	1	no	—
114	#427	—	2	1	no	—
115	#427	—	2	1	no	—
116	C₂×C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
117	#364	—	67	2	no	—
118	#364	—	67	2	no	—
119	#364	—	134	1	no	—
120	#364	—	134	1	no	—
121	C₂²×C₈	C₂²⋊D₆₇	1	1	yes	characteristic
122	C₂²×C₈	C₂²⋊D₆₇	1	1	yes	characteristic
123	C₄⋊Q₈	—	2	2	no	—
124	#125	C₂×Dic₆₇	1	4	yes	—
125	C₂²⋊Q₈	—	4	2	no	—
126	C₄×Q₈	—	4	2	no	—
127	#117	—	2	2	no	—
128	C₂×C₄²	C₂×Dic₆₇	1	1	yes	characteristic
129	M₃₂	—	268	1	no	—
130	M₃₂	—	268	1	no	—
131	M₃₂	—	134	2	no	—
132	M₃₂	—	134	2	no	—
133	C₂×C₁₆	—	268	1	no	—
134	C₂×C₁₆	—	268	1	no	—
135	#107	—	2	2	no	—
136	#107	—	2	2	no	—
137	#106	—	2	2	no	—
138	#106	—	2	2	no	—
139	C₄×C₈	C₂×Dic₆₇	1	1	yes	characteristic
140	C₄×C₈	C₂²⋊D₆₇	1	1	yes	characteristic
141	C₄×C₈	C₂²⋊D₆₇	1	1	yes	characteristic
142	C₄×C₈	C₂²×D₆₇	1	1	yes	characteristic
143	C₂×Q₈	—	4	2	no	—
144	C₂²×C₄	—	2	1	no	—
145	C₂²×C₄	C₂²⋊Dic₆₇	1	1	yes	characteristic
146	C₂×C₈	C₂²⋊Dic₆₇	1	1	yes	characteristic
147	C₂×C₈	C₂²⋊Dic₆₇	1	1	yes	characteristic
148	C₂×C₈	#11761615	1	1	yes	characteristic
149	C₂×C₈	#11761615	1	1	yes	characteristic
150	C₂×C₈	#11761615	1	1	yes	characteristic
151	C₂×C₈	#11761615	1	1	yes	characteristic
152	C₂×C₈	#11761618	1	1	yes	characteristic
153	C₂×C₈	#11761618	1	1	yes	characteristic
154	C₂×C₈	#11761618	1	1	yes	characteristic
155	C₂×C₈	#11761618	1	1	yes	characteristic
156	C₂×C₈	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
157	C₂×C₈	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
158	C₄⋊C₄	—	2	4	no	—
159	C₄⋊C₄	—	4	4	no	—
160	C₄⋊C₄	—	4	2	no	—
161	C₂²⋊C₄	—	4	2	no	—
162	C₄²	—	2	1	no	—
163	C₄²	—	4	2	no	—
164	C₄²	C₂²⋊Dic₆₇	1	1	yes	characteristic
165	C₄²	C₂²×Dic₆₇	1	1	yes	characteristic
166	C₁₆	—	268	2	no	—
167	C₁₆	—	268	2	no	—
168	C₂³	#14523960	1	1	yes	characteristic
169	Q₈	—	8	4	no	—
170	C₂×C₄	—	2	1	no	—
171	C₂×C₄	—	2	1	no	—
172	C₂×C₄	—	4	2	no	—
173	C₂×C₄	—	4	4	no	—
174	C₂×C₄	#14523960	1	1	yes	characteristic
175	C₂×C₄	#14523967	1	1	yes	characteristic
176	C₂×C₄	#14523967	1	1	yes	characteristic
177	C₂×C₄	#14523967	1	1	yes	characteristic
178	C₂×C₄	#14523967	1	1	yes	characteristic
179	C₂×C₄	C₂×C₂²⋊Dic₆₇	1	1	yes	characteristic
180	C₈	#14523967	1	1	yes	characteristic
181	C₈	#14523967	1	1	yes	characteristic
182	C₈	#14523967	1	1	yes	characteristic
183	C₈	#14523967	1	1	yes	characteristic
184	C₈	C₂×#11761615	1	1	yes	characteristic
185	C₈	C₂×#11761615	1	1	yes	characteristic
186	C₈	C₂×#11761618	1	1	yes	characteristic
187	C₈	C₂×#11761618	1	1	yes	characteristic
188	C₂²	—	4	1	no	—
189	C₂²	#16863145	1	1	yes	characteristic
190	C₂²	#16863145	1	1	yes	characteristic
191	C₂²	C₂×#14523960	1	1	yes	characteristic
192	C₄	—	2	2	no	—
193	C₄	—	4	1	no	—
194	C₄	—	8	4	no	—
195	C₄	#16863145	1	1	yes	characteristic
196	C₄	#16863145	1	1	yes	characteristic
197	C₄	C₂×#14523967	1	1	yes	characteristic
198	C₄	C₂×#14523967	1	1	yes	characteristic
199	C₂	—	2	1	no	—
200	C₂	—	2	1	no	—
201	C₂	—	2	1	no	—
202	C₂	C₂×#16863145	1	1	yes	characteristic

	subgroup	G/H
center	C₄	#16863145
commutator	C₄²×C₆₇	C₂²×C₄
Frattini	C₄×C₈	C₂²×D₆₇
Fitting	C₆₇×#3160	C₂
socle	C₂×C₆₇	C₂×#333
prime radical	C₂³×C₆₇	#100
cosocle	C₄×C₈×C₆₇	C₂³
hypercenter	#3160	D₆₇
hypocenter	C₆₇	#11956
LCS	G ▷ C₄²×C₆₇ ▷ C₂×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂×C₄×C₈ ◁ #3160

Complements

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

N	H ≅ G/N	type
3. #22558270	C₂	semidirect
106. C₆₇	#11956	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
#22558270	2
C₆₇×#3160	2
#11956	67