Subgroups — #30771202

Order 17152.

318 conjugacy classes of subgroups.

The subgroup lattice has 226 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22561965	C₂	1	1	yes	maximal, characteristic
2	C₂×#16863174	C₂	1	2	yes	maximal
3	C₄×#14523983	C₂	1	1	yes	maximal, characteristic
4	#22558289	C₂	1	2	yes	maximal
5	C₆₇×#1770	C₂	1	1	yes	maximal, characteristic
6	C₂×#14523983	C₂²	1	1	yes	characteristic
7	C₂×#14523983	C₂²	1	2	yes	—
8	C₂×#14523983	C₂²	1	1	yes	characteristic
9	C₄⋊(C₆₇⋊C₁₆)	C₄	1	2	yes	complemented
10	C₄⋊(C₆₇⋊C₁₆)	C₄	1	2	yes	complemented
11	#16863174	—	2	4	no	—
12	#16863172	C₄	1	2	yes	complemented
13	C₄×C₆₇⋊C₁₆	C₄	1	2	yes	complemented
14	C₂×C₄⋊C₈×C₆₇	C₄	1	1	yes	characteristic
15	C₂×#97×C₆₇	C₂²	1	2	yes	—
16	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
17	C₄×C₄⋊C₄×C₆₇	C₄	1	1	yes	characteristic
18	#336×C₆₇	C₄	1	2	yes	—
19	#14523983	—	2	2	no	—
20	#14523983	—	2	1	no	—
21	#14523983	—	2	1	no	—
22	#14523983	—	2	2	no	—
23	#14523983	D₄	1	2	yes	—
24	#14523983	Q₈	1	2	yes	—
25	C₂×C₆₇⋊C₁₆	C₂×C₄	1	2	yes	—
26	C₂×C₆₇⋊C₁₆	C₂×C₄	1	4	yes	—
27	C₂×C₆₇⋊C₁₆	C₂×C₄	1	2	yes	—
28	C₂²×C₈×C₆₇	C₂×C₄	1	2	yes	—
29	C₂²×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
30	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
31	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
32	C₂×C₄²×C₆₇	C₂×C₄	1	2	yes	—
33	C₂×C₄²×C₆₇	C₂×C₄	1	1	yes	characteristic
34	C₄⋊C₈×C₆₇	—	2	1	no	—
35	C₄⋊C₈×C₆₇	—	2	1	no	—
36	#97×C₆₇	—	2	2	no	—
37	#97×C₆₇	—	2	2	no	—
38	C₄×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
39	C₄×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
40	C₄×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
41	C₄×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
42	#95×C₆₇	—	2	1	no	—
43	C₆₇⋊C₁₆	—	2	2	no	—
44	C₆₇⋊C₁₆	—	2	4	no	—
45	C₆₇⋊C₁₆	C₄⋊C₄	1	4	yes	complemented
46	C₂²×C₄×C₆₇	—	2	2	no	—
47	C₂²×C₄×C₆₇	M₁₆	1	1	yes	characteristic
48	C₂²×C₄×C₆₇	M₁₆	1	1	yes	characteristic
49	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
50	C₂×C₈×C₆₇	—	2	1	no	—
51	C₂×C₈×C₆₇	—	2	1	no	—
52	C₂×C₈×C₆₇	—	2	2	no	—
53	C₂×C₈×C₆₇	—	2	2	no	—
54	C₂×C₈×C₆₇	—	2	2	no	—
55	C₂×C₈×C₆₇	C₄²	1	1	yes	characteristic
56	C₂×C₈×C₆₇	C₄²	1	1	yes	characteristic
57	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
58	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
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	C₄⋊C₄×C₆₇	—	4	2	no	—
64	C₄²×C₆₇	—	2	2	no	—
65	C₄²×C₆₇	—	2	2	no	—
66	C₄²×C₆₇	C₄²	1	1	yes	characteristic
67	C₄²×C₆₇	C₄²	1	1	yes	characteristic
68	C₄²×C₆₇	M₁₆	1	1	yes	characteristic
69	C₄²×C₆₇	M₁₆	1	1	yes	characteristic
70	C₂³×C₆₇	C₂×M₁₆	1	1	yes	characteristic
71	C₂×C₄×C₆₇	—	2	2	no	—
72	C₂×C₄×C₆₇	—	4	1	no	—
73	C₂×C₄×C₆₇	—	4	2	no	—
74	C₂×C₄×C₆₇	—	4	2	no	—
75	C₂×C₄×C₆₇	—	4	2	no	—
76	C₂×C₄×C₆₇	#97	1	1	yes	characteristic
77	C₂×C₄×C₆₇	#97	1	1	yes	characteristic
78	C₂×C₄×C₆₇	#97	1	1	yes	characteristic
79	C₂×C₄×C₆₇	#97	1	1	yes	characteristic
80	C₂×C₄×C₆₇	C₂×C₄²	1	1	yes	characteristic
81	C₂×C₄×C₆₇	C₄×D₄	1	1	yes	characteristic
82	C₂×C₄×C₆₇	C₄×Q₈	1	1	yes	characteristic
83	C₂×C₄×C₆₇	C₂×M₁₆	1	1	yes	characteristic
84	C₂×C₄×C₆₇	#131	1	1	yes	characteristic
85	C₂×C₄×C₆₇	#131	1	1	yes	characteristic
86	C₈×C₆₇	—	2	1	no	—
87	C₈×C₆₇	—	2	1	no	—
88	C₈×C₆₇	—	4	2	no	—
89	C₈×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
90	C₈×C₆₇	#117	1	1	yes	characteristic
91	C₈×C₆₇	C₄×D₄	1	1	yes	characteristic
92	C₈×C₆₇	C₄×Q₈	1	1	yes	characteristic
93	C₂²×C₆₇	—	2	1	no	—
94	C₂²×C₆₇	—	2	1	no	—
95	C₂²×C₆₇	C₂×#97	1	1	yes	characteristic
96	C₂²×C₆₇	#436	1	1	yes	characteristic
97	C₂²×C₆₇	#446	1	1	yes	characteristic
98	C₄×C₆₇	—	2	1	no	—
99	C₄×C₆₇	—	2	1	no	—
100	C₄×C₆₇	—	2	1	no	—
101	C₄×C₆₇	—	8	2	no	—
102	C₄×C₆₇	#347	1	2	yes	complemented
103	C₄×C₆₇	C₄×C₄⋊C₄	1	1	yes	characteristic
104	C₄×C₆₇	#405	1	1	yes	characteristic
105	C₄×C₆₇	#436	1	1	yes	characteristic
106	C₄×C₆₇	#446	1	1	yes	characteristic
107	#12666	—	67	1	no	maximal, Sylow
108	C₂×C₆₇	—	2	1	no	—
109	C₂×C₆₇	—	2	1	no	—
110	C₂×C₆₇	#1770	1	1	yes	characteristic
111	C₂×C₆₇	C₂×#347	1	1	yes	characteristic
112	C₂×C₆₇	#2110	1	1	yes	characteristic
113	#2150	—	67	1	no	—
114	C₂×#347	—	67	2	no	—
115	C₄×M₃₂	—	67	1	no	—
116	#1770	D₆₇	1	1	yes	characteristic
117	#1375	—	67	2	no	—
118	C₆₇	#12666	1	1	yes	characteristic, complemented, Sylow
119	C₂×M₃₂	—	67	1	no	—
120	C₂×M₃₂	—	67	1	no	—
121	C₂×M₃₂	—	67	2	no	—
122	C₂×C₄⋊C₈	Dic₆₇	1	1	yes	characteristic
123	C₂×#97	C₂×D₆₇	1	2	yes	—
124	C₂×C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
125	C₄×C₄⋊C₄	Dic₆₇	1	1	yes	characteristic
126	C₄⋊C₁₆	—	67	2	no	—
127	C₄⋊C₁₆	—	67	2	no	—
128	#347	—	134	4	no	—
129	#346	—	67	2	no	—
130	C₄×C₁₆	—	67	2	no	—
131	#336	Dic₆₇	1	2	yes	—
132	C₂²×C₈	C₂×Dic₆₇	1	2	yes	—
133	C₂²×C₈	C₂×Dic₆₇	1	1	yes	characteristic
134	C₂²×C₈	C₂²×D₆₇	1	1	yes	characteristic
135	C₂×C₄⋊C₄	—	2	1	no	—
136	C₂×C₄²	C₂×Dic₆₇	1	2	yes	—
137	C₂×C₄²	C₂×Dic₆₇	1	1	yes	characteristic
138	M₃₂	—	67	2	no	—
139	M₃₂	—	67	2	no	—
140	M₃₂	—	134	2	no	—
141	M₃₂	—	134	1	no	—
142	M₃₂	—	134	1	no	—
143	M₃₂	—	134	2	no	—
144	C₂×C₁₆	—	67	2	no	—
145	C₂×C₁₆	—	67	2	no	—
146	C₂×C₁₆	—	67	4	no	—
147	C₄⋊C₈	—	2	1	no	—
148	C₄⋊C₈	—	2	1	no	—
149	#97	—	2	2	no	—
150	#97	—	2	2	no	—
151	C₄×C₈	C₄×D₆₇	1	1	yes	characteristic
152	C₄×C₈	C₄×D₆₇	1	1	yes	characteristic
153	C₄×C₈	C₄×D₆₇	1	1	yes	characteristic
154	C₄×C₈	C₄×D₆₇	1	1	yes	characteristic
155	#95	—	2	1	no	—
156	C₂²×C₄	—	2	2	no	—
157	C₂²×C₄	#11761612	1	1	yes	characteristic
158	C₂²×C₄	#11761612	1	1	yes	characteristic
159	C₂²×C₄	C₂²×Dic₆₇	1	1	yes	characteristic
160	C₂×C₈	—	2	1	no	—
161	C₂×C₈	—	2	1	no	—
162	C₂×C₈	—	2	2	no	—
163	C₂×C₈	—	2	2	no	—
164	C₂×C₈	—	2	2	no	—
165	C₂×C₈	C₄×Dic₆₇	1	1	yes	characteristic
166	C₂×C₈	C₄×Dic₆₇	1	1	yes	characteristic
167	C₂×C₈	C₂×C₄×D₆₇	1	1	yes	characteristic
168	C₂×C₈	C₂×C₄×D₆₇	1	1	yes	characteristic
169	C₂×C₈	D₄×D₆₇	1	1	yes	characteristic
170	C₂×C₈	Q₈×D₆₇	1	1	yes	characteristic
171	C₂×C₈	#11761627	1	1	yes	characteristic
172	C₂×C₈	#11761629	1	1	yes	characteristic
173	C₄⋊C₄	—	4	2	no	—
174	C₄²	—	2	2	no	—
175	C₄²	—	2	2	no	—
176	C₄²	C₄×Dic₆₇	1	1	yes	characteristic
177	C₄²	C₄×Dic₆₇	1	1	yes	characteristic
178	C₄²	#11761612	1	1	yes	characteristic
179	C₄²	#11761612	1	1	yes	characteristic
180	C₁₆	—	67	4	no	—
181	C₁₆	—	134	2	no	—
182	C₁₆	—	134	4	no	—
183	C₂³	C₂×#11761612	1	1	yes	characteristic
184	C₂×C₄	—	2	2	no	—
185	C₂×C₄	—	4	1	no	—
186	C₂×C₄	—	4	2	no	—
187	C₂×C₄	—	4	2	no	—
188	C₂×C₄	—	4	2	no	—
189	C₂×C₄	#14523954	1	1	yes	characteristic
190	C₂×C₄	#14523954	1	1	yes	characteristic
191	C₂×C₄	#14523954	1	1	yes	characteristic
192	C₂×C₄	#14523954	1	1	yes	characteristic
193	C₂×C₄	C₂×C₄×Dic₆₇	1	1	yes	characteristic
194	C₂×C₄	D₄×Dic₆₇	1	1	yes	characteristic
195	C₂×C₄	Q₈×Dic₆₇	1	1	yes	characteristic
196	C₂×C₄	C₂×#11761612	1	1	yes	characteristic
197	C₂×C₄	#14524050	1	1	yes	characteristic
198	C₂×C₄	#14524050	1	1	yes	characteristic
199	C₈	—	2	1	no	—
200	C₈	—	2	1	no	—
201	C₈	—	4	2	no	—
202	C₈	C₄⋊C₄×D₆₇	1	1	yes	characteristic
203	C₈	#14524001	1	1	yes	characteristic
204	C₈	#14524005	1	1	yes	characteristic
205	C₈	#14524008	1	1	yes	characteristic
206	C₂²	—	2	1	no	—
207	C₂²	—	2	1	no	—
208	C₂²	C₂×#14523954	1	1	yes	characteristic
209	C₂²	#16863474	1	1	yes	characteristic
210	C₂²	#16863527	1	1	yes	characteristic
211	C₄	—	2	1	no	—
212	C₄	—	2	1	no	—
213	C₄	—	2	1	no	—
214	C₄	—	8	2	no	—
215	C₄	#16863174	1	2	yes	complemented
216	C₄	C₄⋊C₄×Dic₆₇	1	1	yes	characteristic
217	C₄	#16863352	1	1	yes	characteristic
218	C₄	#16863474	1	1	yes	characteristic
219	C₄	#16863527	1	1	yes	characteristic
220	C₂	—	2	1	no	—
221	C₂	—	2	1	no	—
222	C₂	#22560078	1	1	yes	characteristic
223	C₂	C₂×#16863174	1	1	yes	characteristic
224	C₂	#22561814	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×C₄×Dic₆₇
commutator	C₂×C₄×C₆₇	C₂×C₄²
Frattini	C₂²×C₈	C₂²×D₆₇
Fitting	C₆₇×#1770	C₂
socle	C₂²×C₆₇	C₂×#97
prime radical	C₂³×C₆₇	C₂×M₁₆
cosocle	C₂²×C₈×C₆₇	C₂³
hypercenter	#1770	D₆₇
hypocenter	C₆₇	#12666
LCS	G ▷ C₂×C₄×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂×C₄ ◁ #1770

Complements

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

N	H ≅ G/N	type
9. C₄⋊(C₆₇⋊C₁₆)	C₄	semidirect
10. C₄⋊(C₆₇⋊C₁₆)	C₄	semidirect
12. #16863172	C₄	semidirect
13. C₄×C₆₇⋊C₁₆	C₄	semidirect
45. C₆₇⋊C₁₆	C₄⋊C₄	semidirect
102. C₄×C₆₇	#347	semidirect
118. C₆₇	#12666	semidirect
215. C₄	#16863174	semidirect

Maximal subgroups

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

subgroup	index
#22561965	2
C₂×#16863174	2
C₄×#14523983	2
#22558289	2
C₆₇×#1770	2
#12666	67