Subgroups — #30770951

Order 17152.

382 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	#22561830	C₂	1	1	yes	maximal, characteristic
2	#22561813	C₂	1	2	yes	maximal, complemented
3	#22560009	C₂	1	1	yes	maximal, characteristic
4	#22558297	C₂	1	2	yes	maximal, complemented
5	C₂×#504×C₆₇	C₂	1	1	yes	maximal, characteristic
6	C₂×#14523983	C₂²	1	2	yes	—
7	#16863380	C₄	1	1	yes	characteristic
8	#16863370	C₄	1	1	yes	characteristic
9	#16863351	C₂²	1	2	yes	—
10	#16863178	C₄	1	4	yes	complemented
11	#16863173	—	2	4	no	—
12	#16863167	C₄	1	2	yes	—
13	C₂×#131×C₆₇	C₂²	1	1	yes	characteristic
14	#504×C₆₇	—	2	4	no	—
15	C₂×M₃₂×C₆₇	C₄	1	2	yes	complemented
16	C₂×M₃₂×C₆₇	C₂²	1	1	yes	characteristic
17	C₂²×C₁₆×C₆₇	C₄	1	2	yes	complemented
18	C₂²×C₁₆×C₆₇	C₂²	1	1	yes	characteristic
19	C₂×#11761612	C₂×C₄	1	2	yes	—
20	#14524002	C₂×C₄	1	2	yes	—
21	#14523983	—	2	2	no	—
22	#14523983	C₂×C₄	1	4	yes	—
23	C₂×C₆₇⋊C₁₆	—	2	2	no	—
24	#14523962	—	2	1	no	—
25	#14523960	—	2	1	no	—
26	#14523958	—	2	2	no	—
27	#14523953	—	2	2	no	—
28	C₈×Dic₆₇	—	2	2	no	—
29	C₂×#41×C₆₇	C₂×C₄	1	1	yes	characteristic
30	#131×C₆₇	—	2	2	no	—
31	#131×C₆₇	—	2	2	no	—
32	C₂×M₁₆×C₆₇	C₂×C₄	1	1	yes	characteristic
33	C₂×M₁₆×C₆₇	C₂×C₄	1	2	yes	—
34	C₂²×C₈×C₆₇	C₂×C₄	1	2	yes	—
35	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
36	M₃₂×C₆₇	—	2	4	no	—
37	M₃₂×C₆₇	D₄	1	4	yes	—
38	C₂×C₁₆×C₆₇	—	2	2	no	—
39	C₂×C₁₆×C₆₇	—	2	4	no	—
40	C₂×C₁₆×C₆₇	C₂×C₄	1	1	yes	characteristic
41	C₂×C₁₆×C₆₇	C₂×C₄	1	1	yes	characteristic
42	C₂×C₁₆×C₆₇	C₂×C₄	1	2	yes	—
43	#11761612	—	2	2	no	—
44	#11761612	—	2	2	no	—
45	C₂×C₆₇⋊C₈	—	2	2	no	—
46	C₄⋊Dic₆₇	—	2	2	no	—
47	C₂²⋊Dic₆₇	—	2	2	no	—
48	C₄×Dic₆₇	—	2	2	no	—
49	C₆₇⋊C₁₆	—	2	4	no	—
50	#41×C₆₇	—	2	2	no	—
51	#41×C₆₇	—	2	2	no	—
52	C₂×Q₈×C₆₇	M₁₆	1	1	yes	characteristic
53	C₂×D₄×C₆₇	M₁₆	1	2	yes	—
54	C₂×D₄×C₆₇	M₁₆	1	1	yes	characteristic
55	C₂²×C₄×C₆₇	C₄²	1	2	yes	—
56	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
57	M₁₆×C₆₇	—	2	2	no	—
58	M₁₆×C₆₇	—	2	4	no	—
59	C₂×C₈×C₆₇	—	2	2	no	—
60	C₂×C₈×C₆₇	—	2	2	no	—
61	C₂×C₈×C₆₇	C₄²	1	2	yes	—
62	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
63	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
64	C₂×C₈×C₆₇	C₂×D₄	1	2	yes	—
65	C₂×C₈×C₆₇	#41	1	2	yes	—
66	C₁₆×C₆₇	—	2	1	no	—
67	C₁₆×C₆₇	—	2	1	no	—
68	C₁₆×C₆₇	C₂²⋊C₄	1	4	yes	—
69	C₂³×C₆₇	#97	1	2	yes	—
70	C₂³×C₆₇	C₂×M₁₆	1	1	yes	characteristic
71	Q₈×C₆₇	—	2	2	no	—
72	D₄×C₆₇	—	2	4	no	—
73	D₄×C₆₇	—	2	2	no	—
74	C₂×C₄×C₆₇	—	2	2	no	—
75	C₂×C₄×C₆₇	—	2	2	no	—
76	C₂×C₄×C₆₇	#97	1	2	yes	—
77	C₂×C₄×C₆₇	C₂×C₄²	1	1	yes	characteristic
78	C₂×C₄×C₆₇	C₄×D₄	1	2	yes	—
79	C₂×C₄×C₆₇	C₂×M₁₆	1	1	yes	characteristic
80	C₂×C₄×C₆₇	#131	1	2	yes	—
81	C₂×Dic₆₇	—	4	2	no	—
82	C₈×C₆₇	—	2	2	no	—
83	C₈×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
84	C₈×C₆₇	#117	1	1	yes	characteristic
85	C₈×C₆₇	C₄×D₄	1	2	yes	—
86	C₆₇⋊C₈	—	4	2	no	—
87	C₂²×C₆₇	—	2	2	no	—
88	C₂²×C₆₇	—	4	1	no	—
89	C₂²×C₆₇	—	4	2	no	—
90	C₂²×C₆₇	C₂×#97	1	1	yes	characteristic
91	C₂²×C₆₇	#435	1	2	yes	—
92	C₄×C₆₇	—	2	2	no	—
93	C₄×C₆₇	C₄×C₂²⋊C₄	1	1	yes	characteristic
94	C₄×C₆₇	#405	1	1	yes	characteristic
95	C₄×C₆₇	#435	1	2	yes	—
96	Dic₆₇	—	8	2	no	—
97	#12617	—	67	1	no	maximal, Sylow
98	C₂×C₆₇	—	2	2	no	—
99	C₂×C₆₇	—	2	1	no	—
100	C₂×C₆₇	—	4	2	no	—
101	C₂×C₆₇	#1752	1	1	yes	characteristic
102	C₂×#504	D₆₇	1	1	yes	characteristic
103	#2115	—	67	1	no	—
104	#2110	—	67	2	no	—
105	#1757	—	67	1	no	—
106	#1379	—	67	2	no	—
107	C₆₇	#12617	1	1	yes	characteristic, complemented, Sylow
108	C₂×#131	C₂×D₆₇	1	1	yes	characteristic
109	#504	—	2	4	no	—
110	C₂×M₃₂	—	67	2	no	—
111	C₂×M₃₂	Dic₆₇	1	2	yes	complemented
112	C₂×M₃₂	C₂×D₆₇	1	1	yes	characteristic
113	C₂²×C₁₆	Dic₆₇	1	2	yes	complemented
114	C₂²×C₁₆	C₂×D₆₇	1	1	yes	characteristic
115	#413	—	67	1	no	—
116	#410	—	67	1	no	—
117	#405	—	67	2	no	—
118	#349	—	67	4	no	—
119	#347	—	134	4	no	—
120	#344	—	67	2	no	—
121	C₂×#41	C₄×D₆₇	1	1	yes	characteristic
122	#131	—	2	2	no	—
123	#131	—	2	2	no	—
124	C₂×M₁₆	—	67	2	no	—
125	C₂×M₁₆	C₄×D₆₇	1	1	yes	characteristic
126	C₂×M₁₆	C₄×D₆₇	1	2	yes	—
127	C₂²×C₈	C₂×Dic₆₇	1	2	yes	—
128	C₂²×C₈	C₂²×D₆₇	1	1	yes	characteristic
129	#117	—	67	2	no	—
130	M₃₂	—	2	4	no	—
131	M₃₂	—	67	4	no	—
132	M₃₂	—	134	2	no	—
133	M₃₂	C₂²⋊D₆₇	1	4	yes	—
134	C₂×C₁₆	—	2	2	no	—
135	C₂×C₁₆	—	2	4	no	—
136	C₂×C₁₆	—	134	2	no	—
137	C₂×C₁₆	C₂×Dic₆₇	1	1	yes	characteristic
138	C₂×C₁₆	C₂×Dic₆₇	1	1	yes	characteristic
139	C₂×C₁₆	C₂×Dic₆₇	1	2	yes	—
140	#101	—	134	1	no	—
141	#100	—	134	1	no	—
142	#99	—	134	2	no	—
143	#97	—	134	2	no	—
144	C₄×C₈	—	134	2	no	—
145	#41	—	2	2	no	—
146	#41	—	2	2	no	—
147	C₂×Q₈	#11761611	1	1	yes	characteristic
148	C₂×D₄	#11761611	1	2	yes	—
149	C₂×D₄	#11761611	1	1	yes	characteristic
150	C₂²×C₄	C₄×Dic₆₇	1	2	yes	—
151	C₂²×C₄	C₂×C₄×D₆₇	1	1	yes	characteristic
152	M₁₆	—	2	2	no	—
153	M₁₆	—	2	4	no	—
154	M₁₆	—	134	2	no	—
155	M₁₆	—	134	2	no	—
156	C₂×C₈	—	2	2	no	—
157	C₂×C₈	—	2	2	no	—
158	C₂×C₈	—	134	2	no	—
159	C₂×C₈	C₄×Dic₆₇	1	2	yes	—
160	C₂×C₈	C₂×C₄×D₆₇	1	1	yes	characteristic
161	C₂×C₈	C₂²×Dic₆₇	1	1	yes	characteristic
162	C₂×C₈	C₂×C₂²⋊D₆₇	1	2	yes	—
163	C₂×C₈	#11761628	1	2	yes	—
164	C₄⋊C₄	—	134	2	no	—
165	C₂²⋊C₄	—	134	2	no	—
166	C₄²	—	134	2	no	—
167	C₁₆	—	2	1	no	—
168	C₁₆	—	2	1	no	—
169	C₁₆	—	134	4	no	—
170	C₁₆	C₂²⋊Dic₆₇	1	4	yes	—
171	C₂³	#14523953	1	2	yes	—
172	C₂³	C₂×#11761611	1	1	yes	characteristic
173	Q₈	—	2	2	no	—
174	D₄	—	2	4	no	—
175	D₄	—	2	2	no	—
176	C₂×C₄	—	2	2	no	—
177	C₂×C₄	—	2	2	no	—
178	C₂×C₄	—	268	2	no	—
179	C₂×C₄	#14523953	1	2	yes	—
180	C₂×C₄	C₂×C₄×Dic₆₇	1	1	yes	characteristic
181	C₂×C₄	C₄×C₂²⋊D₆₇	1	2	yes	—
182	C₂×C₄	C₂×#11761611	1	1	yes	characteristic
183	C₂×C₄	#14524049	1	2	yes	—
184	C₈	—	2	2	no	—
185	C₈	—	268	2	no	—
186	C₈	C₂×C₂²⋊Dic₆₇	1	1	yes	characteristic
187	C₈	#14524002	1	1	yes	characteristic
188	C₈	C₄×C₂²⋊D₆₇	1	2	yes	—
189	C₂²	—	2	2	no	—
190	C₂²	—	4	1	no	—
191	C₂²	—	4	2	no	—
192	C₂²	C₂×#14523953	1	1	yes	characteristic
193	C₂²	#16863467	1	2	yes	—
194	C₄	—	2	2	no	—
195	C₄	—	536	2	no	—
196	C₄	C₄×C₂²⋊Dic₆₇	1	1	yes	characteristic
197	C₄	#16863351	1	1	yes	characteristic
198	C₄	#16863467	1	2	yes	—
199	C₂	—	2	2	no	—
200	C₂	—	2	1	no	—
201	C₂	—	4	2	no	—
202	C₂	#22559985	1	1	yes	characteristic

	subgroup	G/H
center	C₈	#14524002
commutator	C₂×C₄×C₆₇	C₂×C₄²
Frattini	C₂²×C₈	C₂²×D₆₇
Fitting	C₂×#504×C₆₇	C₂
socle	C₂×C₆₇	#1752
prime radical	C₂×#41×C₆₇	C₂×C₄
cosocle	C₂²×C₈×C₆₇	C₂³
hypercenter	C₂×#504	D₆₇
hypocenter	C₆₇	#12617
LCS	G ▷ C₂×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₈ ◁ C₂×M₃₂ ◁ C₂×#504

Complements

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

N	H ≅ G/N	type
2. #22561813	C₂	semidirect
4. #22558297	C₂	semidirect
10. #16863178	C₄	semidirect
15. C₂×M₃₂×C₆₇	C₄	semidirect
17. C₂²×C₁₆×C₆₇	C₄	semidirect
107. C₆₇	#12617	semidirect
111. C₂×M₃₂	Dic₆₇	semidirect
113. C₂²×C₁₆	Dic₆₇	semidirect

Maximal subgroups

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

subgroup	index
#22561830	2
#22561813	2
#22560009	2
#22558297	2
C₂×#504×C₆₇	2
#12617	67