Subgroups — #30767421

Order 17152.

242 conjugacy classes of subgroups.

The subgroup lattice has 186 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×#14523983	C₂	1	1	yes	maximal, characteristic
2	C₂×#16863149	C₂	1	1	yes	maximal, characteristic
3	#22558282	C₂	1	2	yes	maximal, complemented
4	#22558280	C₂	1	2	yes	maximal, complemented
5	C₆₇×#1564	C₂	1	1	yes	maximal, characteristic
6	C₂×#14523983	C₄	1	1	yes	characteristic
7	C₂×#14523983	C₄	1	1	yes	characteristic
8	C₂×C₄⋊(C₆₇⋊C₈)	—	2	1	no	—
9	#16863172	C₄	1	2	yes	—
10	C₄×C₆₇⋊C₁₆	C₂²	1	2	yes	—
11	#16863149	—	2	1	no	—
12	#16863149	C₂²	1	2	yes	—
13	#431×C₆₇	—	2	1	no	—
14	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
15	#335×C₆₇	C₂²	1	1	yes	characteristic
16	#335×C₆₇	C₂²	1	1	yes	characteristic
17	#334×C₆₇	—	2	1	no	—
18	C₂²×C₆₇⋊C₈	—	4	1	no	—
19	#14523983	—	2	2	no	—
20	#14523983	C₈	1	4	yes	complemented
21	C₂×C₆₇⋊C₁₆	C₂×C₄	1	2	yes	—
22	C₂×C₆₇⋊C₁₆	C₂×C₄	1	2	yes	—
23	C₄⋊(C₆₇⋊C₈)	—	2	2	no	—
24	C₄⋊(C₆₇⋊C₈)	—	4	1	no	—
25	C₂²×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
26	C₂²×C₈×C₆₇	Q₈	1	1	yes	characteristic
27	C₂×C₄²×C₆₇	D₄	1	1	yes	characteristic
28	C₄⋊C₈×C₆₇	—	2	1	no	—
29	C₄⋊C₈×C₆₇	—	2	1	no	—
30	C₂²⋊C₈×C₆₇	—	4	1	no	—
31	C₄×C₈×C₆₇	—	4	1	no	—
32	C₄×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
33	C₄×C₈×C₆₇	D₄	1	1	yes	characteristic
34	C₄×C₈×C₆₇	Q₈	1	1	yes	characteristic
35	C₄×C₈×C₆₇	C₂³	1	1	yes	characteristic
36	C₂×C₆₇⋊C₈	—	4	2	no	—
37	C₂×C₆₇⋊C₈	—	8	2	no	—
38	C₆₇⋊C₁₆	—	2	2	no	—
39	C₆₇⋊C₁₆	C₂×C₈	1	4	yes	complemented
40	C₂²×C₄×C₆₇	C₄⋊C₄	1	1	yes	characteristic
41	C₂²×C₄×C₆₇	D₈	1	1	yes	characteristic
42	C₂²×C₄×C₆₇	Q₁₆	1	1	yes	characteristic
43	C₂×C₈×C₆₇	—	2	2	no	—
44	C₂×C₈×C₆₇	—	4	1	no	—
45	C₂×C₈×C₆₇	—	4	1	no	—
46	C₂×C₈×C₆₇	C₄⋊C₄	1	1	yes	characteristic
47	C₂×C₈×C₆₇	C₄⋊C₄	1	1	yes	characteristic
48	C₂×C₈×C₆₇	C₂×C₈	1	2	yes	—
49	C₂×C₈×C₆₇	M₁₆	1	2	yes	—
50	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
51	C₂×C₈×C₆₇	C₂×Q₈	1	1	yes	characteristic
52	C₄²×C₆₇	C₄⋊C₄	1	1	yes	characteristic
53	C₄²×C₆₇	D₈	1	1	yes	characteristic
54	C₄²×C₆₇	Q₁₆	1	1	yes	characteristic
55	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
56	C₂³×C₆₇	#107	1	1	yes	characteristic
57	C₂×C₄×C₆₇	—	2	1	no	—
58	C₂×C₄×C₆₇	—	2	1	no	—
59	C₂×C₄×C₆₇	—	4	1	no	—
60	C₂×C₄×C₆₇	C₄⋊C₈	1	2	yes	—
61	C₂×C₄×C₆₇	#107	1	1	yes	characteristic
62	C₂×C₄×C₆₇	#107	1	1	yes	characteristic
63	C₂×C₄×C₆₇	#107	1	1	yes	characteristic
64	C₂×C₄×C₆₇	#108	1	2	yes	—
65	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
66	C₂×C₄×C₆₇	C₂×D₈	1	1	yes	characteristic
67	C₂×C₄×C₆₇	C₂×Q₁₆	1	1	yes	characteristic
68	C₈×C₆₇	—	2	1	no	—
69	C₈×C₆₇	—	2	1	no	—
70	C₈×C₆₇	—	8	1	no	—
71	C₈×C₆₇	—	8	1	no	—
72	C₈×C₆₇	C₄⋊C₈	1	2	yes	—
73	C₈×C₆₇	C₂²×C₈	1	1	yes	characteristic
74	C₈×C₆₇	C₂×M₁₆	1	1	yes	characteristic
75	C₆₇⋊C₈	—	8	2	no	—
76	C₂²×C₆₇	—	2	1	no	—
77	C₂²×C₆₇	—	2	1	no	—
78	C₂²×C₆₇	#335	1	2	yes	—
79	C₂²×C₆₇	C₂×#107	1	1	yes	characteristic
80	C₄×C₆₇	—	2	1	no	—
81	C₄×C₆₇	—	2	1	no	—
82	C₄×C₆₇	—	2	1	no	—
83	C₄×C₆₇	#335	1	2	yes	—
84	C₄×C₆₇	C₂×C₄⋊C₈	1	1	yes	characteristic
85	C₄×C₆₇	C₂×#108	1	1	yes	characteristic
86	C₄×C₆₇	#509	1	1	yes	characteristic
87	C₄×C₆₇	#510	1	1	yes	characteristic
88	#12009	—	67	1	no	maximal, Sylow
89	C₂×C₆₇	—	2	2	no	—
90	C₂×C₆₇	C₂×#335	1	1	yes	characteristic
91	C₂×C₆₇	#2159	1	1	yes	characteristic
92	C₂×C₆₇	#2161	1	1	yes	characteristic
93	C₄×M₃₂	—	67	1	no	—
94	#1564	D₆₇	1	1	yes	characteristic
95	C₂×#335	—	67	1	no	—
96	#1372	—	67	2	no	—
97	#1371	—	67	2	no	—
98	C₆₇	#12009	1	1	yes	characteristic, complemented, Sylow
99	C₂×M₃₂	—	67	1	no	—
100	C₂×M₃₂	—	67	1	no	—
101	#431	—	2	1	no	—
102	C₂×C₄⋊C₈	—	134	1	no	—
103	C₂×C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
104	#346	—	67	2	no	—
105	C₄×C₁₆	—	67	2	no	—
106	#335	—	67	2	no	—
107	#335	—	134	1	no	—
108	#335	C₂×D₆₇	1	1	yes	characteristic
109	#335	C₂×D₆₇	1	1	yes	characteristic
110	#334	—	2	1	no	—
111	C₂²×C₈	—	268	1	no	—
112	C₂²×C₈	Dic₁₃₄	1	1	yes	characteristic
113	C₂²×C₈	C₄×D₆₇	1	1	yes	characteristic
114	C₂×C₄²	C₂²⋊D₆₇	1	1	yes	characteristic
115	M₃₂	—	67	4	no	—
116	M₃₂	—	134	2	no	—
117	C₂×C₁₆	—	67	2	no	—
118	C₂×C₁₆	—	67	2	no	—
119	C₄⋊C₈	—	2	1	no	—
120	C₄⋊C₈	—	2	1	no	—
121	C₄⋊C₈	—	268	1	no	—
122	C₄⋊C₈	—	134	2	no	—
123	C₂²⋊C₈	—	4	1	no	—
124	C₄×C₈	—	4	1	no	—
125	C₄×C₈	Dic₁₃₄	1	1	yes	characteristic
126	C₄×C₈	C₄×D₆₇	1	1	yes	characteristic
127	C₄×C₈	C₂²⋊D₆₇	1	1	yes	characteristic
128	C₄×C₈	C₂²×D₆₇	1	1	yes	characteristic
129	C₂²×C₄	Dic₆₇⋊C₄	1	1	yes	characteristic
130	C₂²×C₄	#11761613	1	1	yes	characteristic
131	C₂²×C₄	#11761618	1	1	yes	characteristic
132	C₂×C₈	—	2	2	no	—
133	C₂×C₈	—	4	1	no	—
134	C₂×C₈	—	4	1	no	—
135	C₂×C₈	—	268	2	no	—
136	C₂×C₈	—	536	2	no	—
137	C₂×C₈	Dic₆₇⋊C₄	1	1	yes	characteristic
138	C₂×C₈	Dic₆₇⋊C₄	1	1	yes	characteristic
139	C₂×C₈	C₈×D₆₇	1	2	yes	—
140	C₂×C₈	#11761611	1	2	yes	—
141	C₂×C₈	C₂×C₄×D₆₇	1	1	yes	characteristic
142	C₂×C₈	C₂×Dic₁₃₄	1	1	yes	characteristic
143	C₄²	Dic₆₇⋊C₄	1	1	yes	characteristic
144	C₄²	#11761613	1	1	yes	characteristic
145	C₄²	#11761618	1	1	yes	characteristic
146	C₄²	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
147	C₁₆	—	67	4	no	—
148	C₁₆	—	134	2	no	—
149	C₂³	#14523976	1	1	yes	characteristic
150	C₂×C₄	—	2	1	no	—
151	C₂×C₄	—	2	1	no	—
152	C₂×C₄	—	4	1	no	—
153	C₂×C₄	Dic₆₇⋊C₈	1	2	yes	—
154	C₂×C₄	#14523976	1	1	yes	characteristic
155	C₂×C₄	#14523976	1	1	yes	characteristic
156	C₂×C₄	#14523976	1	1	yes	characteristic
157	C₂×C₄	#14523978	1	2	yes	—
158	C₂×C₄	C₂×Dic₆₇⋊C₄	1	1	yes	characteristic
159	C₂×C₄	C₂×#11761613	1	1	yes	characteristic
160	C₂×C₄	C₂×#11761618	1	1	yes	characteristic
161	C₈	—	2	1	no	—
162	C₈	—	2	1	no	—
163	C₈	—	8	1	no	—
164	C₈	—	8	1	no	—
165	C₈	—	536	2	no	—
166	C₈	Dic₆₇⋊C₈	1	2	yes	—
167	C₈	C₂×C₈×D₆₇	1	1	yes	characteristic
168	C₈	C₂×#11761611	1	1	yes	characteristic
169	C₂²	—	2	1	no	—
170	C₂²	—	2	1	no	—
171	C₂²	#16863148	1	2	yes	—
172	C₂²	C₂×#14523976	1	1	yes	characteristic
173	C₄	—	2	1	no	—
174	C₄	—	2	1	no	—
175	C₄	—	2	1	no	—
176	C₄	#16863148	1	2	yes	—
177	C₄	C₂×Dic₆₇⋊C₈	1	1	yes	characteristic
178	C₄	C₂×#14523978	1	1	yes	characteristic
179	C₄	#16863831	1	1	yes	characteristic
180	C₄	#16863836	1	1	yes	characteristic
181	C₂	—	2	2	no	—
182	C₂	C₂×#16863148	1	1	yes	characteristic
183	C₂	#22561998	1	1	yes	characteristic
184	C₂	#22562003	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×#11761613
commutator	C₈×C₆₇	C₂²×C₈
Frattini	C₄×C₈	C₂²×D₆₇
Fitting	C₆₇×#1564	C₂
socle	C₂²×C₆₇	C₂×#107
prime radical	C₂³×C₆₇	#107
cosocle	C₄×C₈×C₆₇	C₂³
hypercenter	#1564	D₆₇
hypocenter	C₆₇	#12009
LCS	G ▷ C₈×C₆₇ ▷ C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ #1564

Complements

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

N	H ≅ G/N	type
3. #22558282	C₂	semidirect
4. #22558280	C₂	semidirect
20. #14523983	C₈	semidirect
39. C₆₇⋊C₁₆	C₂×C₈	semidirect
98. C₆₇	#12009	semidirect

Maximal subgroups

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

subgroup	index
C₄×#14523983	2
C₂×#16863149	2
#22558282	2
#22558280	2
C₆₇×#1564	2
#12009	67