Subgroups — #30866270

Order 17152.

662 conjugacy classes of subgroups.

The subgroup lattice has 188 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22568847	C₂	1	2	yes	maximal
2	#22560985	C₂	1	8	yes	maximal, complemented
3	#22560251	C₂	1	2	yes	maximal
4	#22560247	C₂	1	2	yes	maximal
5	C₂×#424×C₆₇	C₂	1	1	yes	maximal, characteristic
6	#16863457	—	2	8	no	—
7	C₂×#14523979	C₂²	1	4	yes	—
8	C₂×#14523979	C₂²	1	4	yes	—
9	#16863420	C₂²	1	4	yes	—
10	#16863360	C₂²	1	4	yes	—
11	C₄×#11761612	C₂²	1	4	yes	—
12	#16863161	C₂²	1	4	yes	—
13	#16863161	C₂²	1	4	yes	—
14	C₂²×M₁₆×C₆₇	C₂²	1	1	yes	characteristic
15	C₂³×C₈×C₆₇	C₂²	1	1	yes	characteristic
16	C₂×#117×C₆₇	C₂²	1	1	yes	characteristic
17	#424×C₆₇	C₄	1	8	yes	—
18	C₂×C₄⋊C₈×C₆₇	C₂²	1	4	yes	—
19	C₂×#11761612	C₂³	1	8	yes	—
20	#14523979	—	4	4	no	—
21	#14523979	—	4	4	no	—
22	C₄⋊(C₆₇⋊C₈)	—	2	8	no	—
23	C₂²⋊(C₆₇⋊C₈)	—	2	8	no	—
24	#14523954	—	2	4	no	—
25	C₄×C₆₇⋊C₈	—	2	4	no	—
26	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
27	C₂×M₁₆×C₆₇	—	4	2	no	—
28	C₂×M₁₆×C₆₇	C₂×C₄	1	4	yes	—
29	C₂²×C₈×C₆₇	—	4	2	no	—
30	C₂²×C₈×C₆₇	C₂×C₄	1	4	yes	—
31	C₂²×C₈×C₆₇	C₂³	1	2	yes	—
32	C₂²×C₈×C₆₇	C₂³	1	2	yes	—
33	#117×C₆₇	C₂×C₄	1	4	yes	—
34	#117×C₆₇	D₄	1	4	yes	—
35	C₂×C₄⋊C₄×C₆₇	Q₈	1	2	yes	—
36	C₂×C₂²⋊C₄×C₆₇	Q₈	1	2	yes	—
37	C₂×C₄²×C₆₇	C₂³	1	2	yes	—
38	C₄⋊C₈×C₆₇	C₂×C₄	1	16	yes	—
39	#11761612	—	2	8	no	—
40	#11761612	—	2	8	no	—
41	C₂×C₆₇⋊C₈	—	2	8	no	—
42	C₂⁴×C₆₇	C₂×Q₈	1	1	yes	characteristic
43	C₂²×C₄×C₆₇	—	2	2	no	—
44	C₂²×C₄×C₆₇	—	2	2	no	—
45	C₂²×C₄×C₆₇	C₂²×C₄	1	2	yes	—
46	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
47	C₂²×C₄×C₆₇	C₂×Q₈	1	1	yes	characteristic
48	C₂²×C₄×C₆₇	C₂×Q₈	1	4	yes	—
49	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
50	M₁₆×C₆₇	—	4	4	no	—
51	C₂×C₈×C₆₇	—	2	4	no	—
52	C₂×C₈×C₆₇	—	2	4	no	—
53	C₂×C₈×C₆₇	—	4	4	no	—
54	C₂×C₈×C₆₇	C₂²×C₄	1	4	yes	—
55	C₂×C₈×C₆₇	C₂²×C₄	1	4	yes	—
56	C₄⋊C₄×C₆₇	C₄⋊C₄	1	8	yes	—
57	C₂²⋊C₄×C₆₇	C₄⋊C₄	1	8	yes	—
58	C₄²×C₆₇	C₂²×C₄	1	4	yes	—
59	C₄²×C₆₇	C₂×D₄	1	4	yes	—
60	C₂³×C₆₇	—	2	2	no	—
61	C₂³×C₆₇	—	4	2	no	—
62	C₂³×C₆₇	C₂×C₄⋊C₄	1	2	yes	—
63	C₂³×C₆₇	C₂²×Q₈	1	1	yes	characteristic
64	C₂×C₄×C₆₇	—	2	2	no	—
65	C₂×C₄×C₆₇	—	2	2	no	—
66	C₂×C₄×C₆₇	—	2	2	no	—
67	C₂×C₄×C₆₇	—	2	4	no	—
68	C₂×C₄×C₆₇	—	4	4	no	—
69	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	8	yes	—
70	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	2	yes	—
71	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
72	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
73	C₂×C₄×C₆₇	H(5,𝔽₂)	1	2	yes	—
74	C₂×C₄×C₆₇	2₋¹⁺⁴	1	2	yes	—
75	C₈×C₆₇	—	2	4	no	—
76	C₈×C₆₇	—	2	4	no	—
77	C₆₇⋊C₈	—	4	8	no	—
78	C₂²×C₆₇	—	2	2	no	—
79	C₂²×C₆₇	—	2	2	no	—
80	C₂²×C₆₇	—	2	4	no	—
81	C₂²×C₆₇	—	4	4	no	—
82	C₂²×C₆₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
83	C₂²×C₆₇	#544	1	2	yes	—
84	C₄×C₆₇	—	2	2	no	—
85	C₄×C₆₇	—	2	8	no	—
86	C₄×C₆₇	#518	1	1	yes	characteristic
87	C₄×C₆₇	#519	1	1	yes	characteristic
88	C₄×C₆₇	#537	1	2	yes	—
89	#21971	—	67	1	no	maximal, Sylow
90	C₂×C₆₇	—	2	2	no	—
91	C₂×C₆₇	—	4	2	no	—
92	C₂×C₆₇	#2316	1	1	yes	characteristic
93	C₂×C₆₇	#2916	1	2	yes	—
94	#2921	—	67	2	no	—
95	C₂×#424	D₆₇	1	1	yes	characteristic
96	#1950	—	67	8	no	—
97	#1811	—	67	2	no	—
98	#1810	—	67	2	no	—
99	C₆₇	#21971	1	1	yes	characteristic, complemented, Sylow
100	C₂²×M₁₆	C₂×D₆₇	1	1	yes	characteristic
101	C₂³×C₈	C₂×D₆₇	1	1	yes	characteristic
102	C₂×#117	C₂×D₆₇	1	1	yes	characteristic
103	#433	—	134	8	no	—
104	C₂×#108	—	67	4	no	—
105	C₂×#108	—	67	4	no	—
106	#424	Dic₆₇	1	8	yes	—
107	#423	—	67	4	no	—
108	C₂×C₄⋊C₈	C₂×D₆₇	1	4	yes	—
109	#407	—	67	4	no	—
110	C₄×M₁₆	—	67	4	no	—
111	#341	—	67	4	no	—
112	#341	—	67	4	no	—
113	C₂³×C₄	C₂²×D₆₇	1	1	yes	characteristic
114	C₂×M₁₆	—	4	2	no	—
115	C₂×M₁₆	—	67	8	no	—
116	C₂×M₁₆	C₂×Dic₆₇	1	4	yes	—
117	C₂²×C₈	—	4	2	no	—
118	C₂²×C₈	C₂×Dic₆₇	1	4	yes	—
119	C₂²×C₈	C₂²×D₆₇	1	2	yes	—
120	C₂²×C₈	C₂²×D₆₇	1	2	yes	—
121	#117	D₂₆₈	1	4	yes	—
122	#117	C₂×Dic₆₇	1	4	yes	—
123	C₂×C₄⋊C₄	Dic₁₃₄	1	2	yes	—
124	C₂×C₂²⋊C₄	Dic₁₃₄	1	2	yes	—
125	C₂×C₄²	C₂²×D₆₇	1	2	yes	—
126	#108	—	268	4	no	—
127	#108	—	268	4	no	—
128	C₄⋊C₈	—	134	8	no	—
129	C₄⋊C₈	C₂×Dic₆₇	1	16	yes	—
130	C₂²⋊C₈	—	134	8	no	—
131	#97	—	134	4	no	—
132	C₄×C₈	—	134	4	no	—
133	C₂⁴	C₂×Dic₁₃₄	1	1	yes	characteristic
134	C₂²×C₄	—	2	2	no	—
135	C₂²×C₄	—	2	2	no	—
136	C₂²×C₄	C₂²×Dic₆₇	1	2	yes	—
137	C₂²×C₄	C₂×D₂₆₈	1	2	yes	—
138	C₂²×C₄	C₂×Dic₁₃₄	1	1	yes	characteristic
139	C₂²×C₄	C₂×Dic₁₃₄	1	4	yes	—
140	C₂²×C₄	C₂³×D₆₇	1	1	yes	characteristic
141	M₁₆	—	4	4	no	—
142	M₁₆	—	134	8	no	—
143	M₁₆	—	134	8	no	—
144	C₂×C₈	—	2	4	no	—
145	C₂×C₈	—	2	4	no	—
146	C₂×C₈	—	4	4	no	—
147	C₂×C₈	—	134	8	no	—
148	C₂×C₈	C₂²×Dic₆₇	1	4	yes	—
149	C₂×C₈	C₂²×Dic₆₇	1	4	yes	—
150	C₄⋊C₄	C₄⋊Dic₆₇	1	8	yes	—
151	C₂²⋊C₄	C₄⋊Dic₆₇	1	8	yes	—
152	C₄²	C₂²×Dic₆₇	1	4	yes	—
153	C₄²	C₂×D₂₆₈	1	4	yes	—
154	C₂³	—	2	2	no	—
155	C₂³	—	4	2	no	—
156	C₂³	C₂×C₄⋊Dic₆₇	1	2	yes	—
157	C₂³	C₂²×Dic₁₃₄	1	1	yes	characteristic
158	C₂×C₄	—	2	2	no	—
159	C₂×C₄	—	2	2	no	—
160	C₂×C₄	—	2	2	no	—
161	C₂×C₄	—	2	4	no	—
162	C₂×C₄	—	4	4	no	—
163	C₂×C₄	C₂×C₄⋊Dic₆₇	1	8	yes	—
164	C₂×C₄	C₂×C₄⋊Dic₆₇	1	2	yes	—
165	C₂×C₄	C₂³×Dic₆₇	1	1	yes	characteristic
166	C₂×C₄	C₂²×D₂₆₈	1	1	yes	characteristic
167	C₂×C₄	#14524088	1	2	yes	—
168	C₂×C₄	#14524091	1	2	yes	—
169	C₈	—	2	4	no	—
170	C₈	—	2	4	no	—
171	C₈	—	268	8	no	—
172	C₂²	—	2	2	no	—
173	C₂²	—	2	2	no	—
174	C₂²	—	2	4	no	—
175	C₂²	—	4	4	no	—
176	C₂²	C₂²×C₄⋊Dic₆₇	1	1	yes	characteristic
177	C₂²	#16864027	1	2	yes	—
178	C₄	—	2	2	no	—
179	C₄	—	2	8	no	—
180	C₄	#16863871	1	1	yes	characteristic
181	C₄	#16863874	1	1	yes	characteristic
182	C₄	#16863977	1	2	yes	—
183	C₂	—	2	2	no	—
184	C₂	—	4	2	no	—
185	C₂	#22562827	1	1	yes	characteristic
186	C₂	#22568820	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₂²×D₂₆₈
commutator	C₂×C₄×C₆₇	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₆₇
Fitting	C₂×#424×C₆₇	C₂
socle	C₂²×C₆₇	C₂²×C₄⋊C₄
prime radical	C₂⁴×C₆₇	C₂×Q₈
cosocle	C₂²×C₄×C₆₇	C₂⁴
hypercenter	C₂×#424	D₆₇
hypocenter	C₆₇	#21971
LCS	G ▷ C₂×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×#117 ◁ C₂×#424

Complements

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

N	H ≅ G/N	type
2. #22560985	C₂	semidirect
99. C₆₇	#21971	semidirect

Maximal subgroups

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

subgroup	index
#22568847	2
#22560985	2
#22560251	2
#22560247	2
C₂×#424×C₆₇	2
#21971	67