Subgroups — #30772249

Order 17152.

342 conjugacy classes of subgroups.

The subgroup lattice has 210 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×#16863216	C₂	1	2	yes	maximal
3	#22561964	C₂	1	2	yes	maximal
4	#22559895	C₂	1	1	yes	maximal, characteristic
5	C₂×#364×C₆₇	C₂	1	1	yes	maximal, characteristic
6	C₂×#14523983	C₂²	1	2	yes	—
7	C₂×#14523954	C₄	1	1	yes	characteristic
8	C₂×#14523953	C₄	1	1	yes	characteristic
9	C₂×#14523953	C₄	1	1	yes	characteristic
10	C₂×C₈×Dic₆₇	C₂²	1	1	yes	characteristic
11	C₂×C₈×Dic₆₇	C₂²	1	1	yes	characteristic
12	#16863242	C₄	1	1	yes	characteristic
13	#16863217	—	2	2	no	—
14	#16863216	—	2	4	no	—
15	Dic₆₇⋊C₁₆	—	2	4	no	—
16	C₂×M₃₂×C₆₇	C₂²	1	2	yes	—
17	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
18	#364×C₆₇	—	2	2	no	—
19	C₂²×C₆₇⋊C₈	C₂×C₄	1	1	yes	characteristic
20	C₂²×C₆₇⋊C₈	C₂×C₄	1	1	yes	characteristic
21	C₂×C₄×Dic₆₇	C₂×C₄	1	1	yes	characteristic
22	C₂×C₄×Dic₆₇	C₂×C₄	1	1	yes	characteristic
23	#14523983	—	2	2	no	—
24	#14523983	—	2	2	no	—
25	C₂×C₆₇⋊C₁₆	—	2	2	no	—
26	#14523954	—	2	2	no	—
27	#14523953	—	2	2	no	—
28	#14523953	C₈	1	4	yes	—
29	C₈×Dic₆₇	D₄	1	2	yes	—
30	C₈×Dic₆₇	D₄	1	2	yes	—
31	C₈×Dic₆₇	Q₈	1	2	yes	—
32	C₈×Dic₆₇	Q₈	1	2	yes	—
33	C₆₇⋊#95	—	2	2	no	—
34	C₂²×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
35	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
36	C₂×C₄²×C₆₇	C₂×C₄	1	1	yes	characteristic
37	M₃₂×C₆₇	—	2	2	no	—
38	M₃₂×C₆₇	—	2	2	no	—
39	C₂×C₁₆×C₆₇	—	2	2	no	—
40	C₄×C₈×C₆₇	D₄	1	2	yes	—
41	C₄×C₈×C₆₇	Q₈	1	2	yes	—
42	C₂²×Dic₆₇	—	2	1	no	—
43	C₂²×Dic₆₇	M₁₆	1	2	yes	—
44	C₂×C₆₇⋊C₈	—	2	1	no	—
45	C₂×C₆₇⋊C₈	—	2	1	no	—
46	C₂×C₆₇⋊C₈	C₄⋊C₄	1	2	yes	—
47	C₂×C₆₇⋊C₈	C₄⋊C₄	1	2	yes	—
48	C₂×C₆₇⋊C₈	C₂×C₈	1	2	yes	—
49	C₂×C₆₇⋊C₈	M₁₆	1	2	yes	—
50	C₄×Dic₆₇	—	2	1	no	—
51	C₄×Dic₆₇	C₄⋊C₄	1	2	yes	—
52	C₄×Dic₆₇	C₄⋊C₄	1	2	yes	—
53	C₄×Dic₆₇	C₂×C₈	1	2	yes	—
54	C₆₇⋊C₁₆	—	4	2	no	—
55	C₂²×C₄×C₆₇	—	2	1	no	—
56	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
57	C₂×C₈×C₆₇	C₄⋊C₄	1	2	yes	—
58	C₂×C₈×C₆₇	C₂×C₈	1	2	yes	—
59	C₂×C₈×C₆₇	M₁₆	1	2	yes	—
60	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
61	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
62	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
63	C₂×C₈×C₆₇	C₂×Q₈	1	1	yes	characteristic
64	C₂×C₈×C₆₇	C₂×Q₈	1	1	yes	characteristic
65	C₂×C₈×C₆₇	C₂×Q₈	1	1	yes	characteristic
66	C₄²×C₆₇	—	2	1	no	—
67	C₄²×C₆₇	C₄⋊C₄	1	2	yes	—
68	C₁₆×C₆₇	—	4	2	no	—
69	C₂³×C₆₇	C₂×M₁₆	1	1	yes	characteristic
70	C₂×C₄×C₆₇	—	2	2	no	—
71	C₂×C₄×C₆₇	—	4	2	no	—
72	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
73	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
74	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
75	C₂×C₄×C₆₇	C₂²×C₈	1	1	yes	characteristic
76	C₂×C₄×C₆₇	C₂×M₁₆	1	1	yes	characteristic
77	C₂×C₄×C₆₇	C₂×M₁₆	1	1	yes	characteristic
78	C₂×Dic₆₇	—	2	2	no	—
79	C₂×Dic₆₇	—	4	2	no	—
80	C₂×Dic₆₇	—	4	2	no	—
81	C₂×Dic₆₇	C₄⋊C₈	1	4	yes	—
82	C₈×C₆₇	C₄⋊C₈	1	4	yes	—
83	C₈×C₆₇	#127	1	1	yes	characteristic
84	C₈×C₆₇	C₄⋊Q₈	1	1	yes	characteristic
85	C₈×C₆₇	C₄⋊Q₈	1	1	yes	characteristic
86	C₈×C₆₇	C₄⋊Q₈	1	1	yes	characteristic
87	C₆₇⋊C₈	—	2	2	no	—
88	C₆₇⋊C₈	C₄⋊C₈	1	4	yes	—
89	C₂²×C₆₇	—	2	2	no	—
90	C₂²×C₆₇	C₂×C₄⋊C₈	1	1	yes	characteristic
91	C₂²×C₆₇	#423	1	1	yes	characteristic
92	C₂²×C₆₇	#423	1	1	yes	characteristic
93	C₄×C₆₇	—	4	2	no	—
94	C₄×C₆₇	#384	1	1	yes	characteristic
95	C₄×C₆₇	C₂×C₄⋊C₈	1	1	yes	characteristic
96	C₄×C₆₇	C₂×C₄⋊C₈	1	1	yes	characteristic
97	C₄×C₆₇	#423	1	1	yes	characteristic
98	Dic₆₇	—	2	4	no	—
99	Dic₆₇	—	4	2	no	—
100	#12901	—	67	1	no	maximal, Sylow
101	C₂×C₆₇	—	2	1	no	—
102	C₂×C₆₇	—	2	1	no	—
103	C₂×C₆₇	#1842	1	1	yes	characteristic
104	C₂×C₆₇	#2153	1	2	yes	—
105	C₂×#364	—	67	2	no	—
106	C₂×#364	—	67	1	no	—
107	C₂×#364	D₆₇	1	1	yes	characteristic
108	#2150	—	67	2	no	—
109	#1726	—	67	1	no	—
110	C₆₇	#12901	1	1	yes	characteristic, complemented, Sylow
111	C₂×M₃₂	—	67	2	no	—
112	C₂×M₃₂	C₂×D₆₇	1	2	yes	—
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	—	67	1	no	—
118	C₂×C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
119	#376	—	67	1	no	—
120	#364	—	2	2	no	—
121	#364	—	134	4	no	—
122	#364	—	134	2	no	—
123	C₄⋊C₁₆	—	134	4	no	—
124	C₂²×C₈	—	67	1	no	—
125	C₂²×C₈	—	67	1	no	—
126	C₂²×C₈	C₄×D₆₇	1	1	yes	characteristic
127	C₂²×C₈	C₂²×D₆₇	1	1	yes	characteristic
128	C₂×C₄²	—	67	1	no	—
129	C₂×C₄²	—	67	1	no	—
130	C₂×C₄²	C₄×D₆₇	1	1	yes	characteristic
131	M₃₂	—	2	2	no	—
132	M₃₂	—	2	2	no	—
133	M₃₂	—	134	2	no	—
134	M₃₂	—	134	2	no	—
135	C₂×C₁₆	—	2	2	no	—
136	C₂×C₁₆	—	134	2	no	—
137	#97	—	67	4	no	—
138	#97	—	134	2	no	—
139	#97	—	134	2	no	—
140	C₄×C₈	—	67	2	no	—
141	C₄×C₈	—	67	2	no	—
142	C₄×C₈	—	67	2	no	—
143	C₄×C₈	—	67	2	no	—
144	C₄×C₈	Dic₁₃₄	1	2	yes	—
145	C₄×C₈	C₂²⋊D₆₇	1	2	yes	—
146	#95	—	134	2	no	—
147	C₂²×C₄	—	2	1	no	—
148	C₂²×C₄	—	67	2	no	—
149	C₂²×C₄	—	134	1	no	—
150	C₂²×C₄	C₂×C₄×D₆₇	1	1	yes	characteristic
151	C₂×C₈	—	67	2	no	—
152	C₂×C₈	—	67	2	no	—
153	C₂×C₈	—	67	2	no	—
154	C₂×C₈	—	67	2	no	—
155	C₂×C₈	—	134	1	no	—
156	C₂×C₈	—	134	1	no	—
157	C₂×C₈	Dic₆₇⋊C₄	1	2	yes	—
158	C₂×C₈	C₈×D₆₇	1	2	yes	—
159	C₂×C₈	#11761611	1	2	yes	—
160	C₂×C₈	D₄×D₆₇	1	1	yes	characteristic
161	C₂×C₈	D₄×D₆₇	1	1	yes	characteristic
162	C₂×C₈	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
163	C₂×C₈	Q₈×D₆₇	1	1	yes	characteristic
164	C₂×C₈	Q₈×D₆₇	1	1	yes	characteristic
165	C₂×C₈	C₂×Dic₁₃₄	1	1	yes	characteristic
166	C₄²	—	2	1	no	—
167	C₄²	—	67	2	no	—
168	C₄²	—	67	2	no	—
169	C₄²	—	67	2	no	—
170	C₄²	—	134	1	no	—
171	C₄²	Dic₆₇⋊C₄	1	2	yes	—
172	C₁₆	—	4	2	no	—
173	C₁₆	—	268	2	no	—
174	C₂³	M₁₆×D₆₇	1	1	yes	characteristic
175	C₂×C₄	—	2	2	no	—
176	C₂×C₄	—	4	2	no	—
177	C₂×C₄	—	268	2	no	—
178	C₂×C₄	—	268	2	no	—
179	C₂×C₄	—	67	4	no	—
180	C₂×C₄	—	134	2	no	—
181	C₂×C₄	C₄⋊C₄×D₆₇	1	1	yes	characteristic
182	C₂×C₄	C₄⋊C₄×D₆₇	1	1	yes	characteristic
183	C₂×C₄	C₂×Dic₆₇⋊C₄	1	1	yes	characteristic
184	C₂×C₄	C₂×C₈×D₆₇	1	1	yes	characteristic
185	C₂×C₄	M₁₆×D₆₇	1	1	yes	characteristic
186	C₂×C₄	C₂×#11761611	1	1	yes	characteristic
187	C₈	—	67	4	no	—
188	C₈	—	134	2	no	—
189	C₈	Dic₆₇⋊C₈	1	4	yes	—
190	C₈	#14524039	1	1	yes	characteristic
191	C₈	#14524041	1	1	yes	characteristic
192	C₈	#14524042	1	1	yes	characteristic
193	C₈	#14524042	1	1	yes	characteristic
194	C₂²	—	2	2	no	—
195	C₂²	C₄⋊C₈×D₆₇	1	1	yes	characteristic
196	C₂²	#16863418	1	1	yes	characteristic
197	C₂²	#16863419	1	1	yes	characteristic
198	C₄	—	4	2	no	—
199	C₄	—	268	2	no	—
200	C₄	—	134	4	no	—
201	C₄	#16863265	1	1	yes	characteristic
202	C₄	C₄⋊C₈×D₆₇	1	1	yes	characteristic
203	C₄	C₂×Dic₆₇⋊C₈	1	1	yes	characteristic
204	C₄	#16863418	1	1	yes	characteristic
205	C₂	—	2	1	no	—
206	C₂	—	2	1	no	—
207	C₂	#22560360	1	1	yes	characteristic
208	C₂	#22561973	1	2	yes	—

	subgroup	G/H
center	C₂×C₈	D₄×D₆₇
commutator	C₂×C₄×C₆₇	C₂²×C₈
Frattini	C₂²×C₈	C₂²×D₆₇
Fitting	C₂×#364×C₆₇	C₂
socle	C₂²×C₆₇	C₂×C₄⋊C₈
prime radical	C₂³×C₆₇	C₂×M₁₆
cosocle	C₂²×C₈×C₆₇	C₂³
hypercenter	C₂×#364	D₆₇
hypocenter	C₆₇	#12901
LCS	G ▷ C₂×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂×C₈ ◁ C₂²×C₈ ◁ C₂×#364

Complements

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

N	H ≅ G/N	type
110. C₆₇	#12901	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
C₂×#16863216	2
#22561964	2
#22559895	2
C₂×#364×C₆₇	2
#12901	67