Subgroups — #30767433

Order 17152.

250 conjugacy classes of subgroups.

The subgroup lattice has 202 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22562012	C₂	1	1	yes	maximal, characteristic
2	C₂×C₈×C₆₇⋊C₈	C₂	1	1	yes	maximal, characteristic
3	#22558278	C₂	1	2	yes	maximal, complemented
4	#22558272	C₂	1	2	yes	maximal, complemented
5	C₆₇×#2163	C₂	1	1	yes	maximal, characteristic
6	C₂×C₄×C₆₇⋊C₈	C₄	1	1	yes	characteristic
7	C₂×C₄×C₆₇⋊C₈	C₄	1	1	yes	characteristic
8	C₄⋊(C₆₇⋊C₁₆)	C₂²	1	2	yes	—
9	C₂²⋊(C₆₇⋊C₁₆)	—	2	1	no	—
10	#16863172	—	2	1	no	—
11	C₈×C₆₇⋊C₈	C₄	1	1	yes	characteristic
12	C₈×C₆₇⋊C₈	C₄	1	1	yes	characteristic
13	C₈×C₆₇⋊C₈	C₂²	1	1	yes	characteristic
14	C₈×C₆₇⋊C₈	C₂²	1	1	yes	characteristic
15	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
16	C₄⋊C₁₆×C₆₇	C₂²	1	2	yes	—
17	C₂²⋊C₁₆×C₆₇	—	2	1	no	—
18	#346×C₆₇	—	2	1	no	—
19	C₂²×C₆₇⋊C₈	—	2	1	no	—
20	C₂²×C₆₇⋊C₈	C₈	1	1	yes	characteristic
21	C₂²×C₆₇⋊C₈	C₈	1	1	yes	characteristic
22	C₂×C₆₇⋊C₁₆	—	2	2	no	—
23	C₄×C₆₇⋊C₈	—	2	1	no	—
24	C₄×C₆₇⋊C₈	C₈	1	1	yes	characteristic
25	C₄×C₆₇⋊C₈	C₈	1	1	yes	characteristic
26	C₄×C₆₇⋊C₈	C₂×C₄	1	1	yes	characteristic
27	C₄×C₆₇⋊C₈	C₂×C₄	1	1	yes	characteristic
28	C₄×C₆₇⋊C₈	C₂×C₄	1	1	yes	characteristic
29	C₄×C₆₇⋊C₈	C₂×C₄	1	1	yes	characteristic
30	C₂²×C₈×C₆₇	D₄	1	1	yes	characteristic
31	C₂²×C₈×C₆₇	Q₈	1	1	yes	characteristic
32	C₂×C₄²×C₆₇	C₂×C₄	1	1	yes	characteristic
33	C₂×C₁₆×C₆₇	—	2	2	no	—
34	C₄×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
35	C₄×C₈×C₆₇	D₄	1	1	yes	characteristic
36	C₄×C₈×C₆₇	Q₈	1	1	yes	characteristic
37	C₄×C₈×C₆₇	C₂³	1	1	yes	characteristic
38	C₂×C₆₇⋊C₈	—	2	4	no	—
39	C₂×C₆₇⋊C₈	—	2	1	no	—
40	C₂×C₆₇⋊C₈	—	2	1	no	—
41	C₂×C₆₇⋊C₈	—	2	2	no	—
42	C₂×C₆₇⋊C₈	—	2	2	no	—
43	C₂×C₆₇⋊C₈	C₂×C₈	1	1	yes	characteristic
44	C₂×C₆₇⋊C₈	C₂×C₈	1	1	yes	characteristic
45	C₂×C₆₇⋊C₈	C₂×C₈	1	1	yes	characteristic
46	C₂×C₆₇⋊C₈	C₂×C₈	1	1	yes	characteristic
47	C₆₇⋊C₁₆	—	4	2	no	—
48	C₂²×C₄×C₆₇	C₄⋊C₄	1	1	yes	characteristic
49	C₂²×C₄×C₆₇	C₂×C₈	1	1	yes	characteristic
50	C₂²×C₄×C₆₇	M₁₆	1	1	yes	characteristic
51	C₂×C₈×C₆₇	—	2	2	no	—
52	C₂×C₈×C₆₇	—	2	2	no	—
53	C₂×C₈×C₆₇	C₄⋊C₄	1	1	yes	characteristic
54	C₂×C₈×C₆₇	C₄⋊C₄	1	1	yes	characteristic
55	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
56	C₂×C₈×C₆₇	C₂×Q₈	1	1	yes	characteristic
57	C₄²×C₆₇	C₄⋊C₄	1	1	yes	characteristic
58	C₄²×C₆₇	C₂×C₈	1	1	yes	characteristic
59	C₄²×C₆₇	M₁₆	1	1	yes	characteristic
60	C₄²×C₆₇	C₂²×C₄	1	1	yes	characteristic
61	C₁₆×C₆₇	—	4	2	no	—
62	C₂³×C₆₇	C₄⋊C₈	1	1	yes	characteristic
63	C₂×C₄×C₆₇	—	2	2	no	—
64	C₂×C₄×C₆₇	—	2	1	no	—
65	C₂×C₄×C₆₇	—	2	1	no	—
66	C₂×C₄×C₆₇	—	2	1	no	—
67	C₂×C₄×C₆₇	—	2	1	no	—
68	C₂×C₄×C₆₇	C₄⋊C₈	1	1	yes	characteristic
69	C₂×C₄×C₆₇	C₄⋊C₈	1	1	yes	characteristic
70	C₂×C₄×C₆₇	C₄⋊C₈	1	1	yes	characteristic
71	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
72	C₂×C₄×C₆₇	C₂²×C₈	1	1	yes	characteristic
73	C₂×C₄×C₆₇	C₂×M₁₆	1	1	yes	characteristic
74	C₈×C₆₇	—	2	1	no	—
75	C₈×C₆₇	—	2	1	no	—
76	C₈×C₆₇	#108	1	2	yes	—
77	C₈×C₆₇	#135	1	2	yes	—
78	C₆₇⋊C₈	—	2	2	no	—
79	C₆₇⋊C₈	—	2	2	no	—
80	C₆₇⋊C₈	—	2	1	no	—
81	C₆₇⋊C₈	—	2	1	no	—
82	C₆₇⋊C₈	M₃₂	1	2	yes	—
83	C₆₇⋊C₈	M₃₂	1	2	yes	—
84	C₂²×C₆₇	—	2	1	no	—
85	C₂²×C₆₇	—	2	1	no	—
86	C₂²×C₆₇	#364	1	2	yes	—
87	C₂²×C₆₇	C₂×C₄⋊C₈	1	1	yes	characteristic
88	C₄×C₆₇	—	2	1	no	—
89	C₄×C₆₇	—	2	1	no	—
90	C₄×C₆₇	—	2	1	no	—
91	C₄×C₆₇	—	2	1	no	—
92	C₄×C₆₇	#427	1	1	yes	characteristic
93	C₄×C₆₇	C₂×#108	1	1	yes	characteristic
94	C₄×C₆₇	C₂×M₃₂	1	1	yes	characteristic
95	C₄×C₆₇	C₂×M₃₂	1	1	yes	characteristic
96	#12013	—	67	1	no	maximal, Sylow
97	C₂×C₆₇	—	2	2	no	—
98	C₂×C₆₇	#1564	1	1	yes	characteristic
99	C₂×C₆₇	#2150	1	1	yes	characteristic
100	C₂×C₆₇	C₂×#364	1	1	yes	characteristic
101	#2163	—	67	1	no	—
102	#2163	D₆₇	1	1	yes	characteristic
103	C₂×C₈²	—	67	1	no	—
104	#1370	—	67	2	no	—
105	#1367	—	67	2	no	—
106	C₆₇	#12013	1	1	yes	characteristic, complemented, Sylow
107	C₂×C₄×C₈	—	67	1	no	—
108	C₂×C₄×C₈	—	67	1	no	—
109	C₂×C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
110	C₄⋊C₁₆	—	67	2	no	—
111	C₄⋊C₁₆	C₂×D₆₇	1	2	yes	—
112	C₂²⋊C₁₆	—	2	1	no	—
113	C₂²⋊C₁₆	—	134	1	no	—
114	#346	—	2	1	no	—
115	#346	—	134	1	no	—
116	C₈²	—	67	1	no	—
117	C₈²	—	67	1	no	—
118	C₈²	—	67	1	no	—
119	C₈²	—	67	1	no	—
120	C₂²×C₈	—	67	1	no	—
121	C₂²×C₈	—	67	1	no	—
122	C₂²×C₈	—	134	1	no	—
123	C₂²×C₈	Dic₁₃₄	1	1	yes	characteristic
124	C₂²×C₈	C₂²⋊D₆₇	1	1	yes	characteristic
125	C₂×C₄²	C₄×D₆₇	1	1	yes	characteristic
126	C₂×C₁₆	—	2	2	no	—
127	C₂×C₁₆	—	134	2	no	—
128	C₄×C₈	—	67	1	no	—
129	C₄×C₈	—	67	1	no	—
130	C₄×C₈	—	67	1	no	—
131	C₄×C₈	—	67	1	no	—
132	C₄×C₈	—	67	1	no	—
133	C₄×C₈	—	67	1	no	—
134	C₄×C₈	—	134	1	no	—
135	C₄×C₈	Dic₁₃₄	1	1	yes	characteristic
136	C₄×C₈	C₄×D₆₇	1	1	yes	characteristic
137	C₄×C₈	C₂²⋊D₆₇	1	1	yes	characteristic
138	C₄×C₈	C₂²×D₆₇	1	1	yes	characteristic
139	C₂²×C₄	Dic₆₇⋊C₄	1	1	yes	characteristic
140	C₂²×C₄	C₈×D₆₇	1	1	yes	characteristic
141	C₂²×C₄	#11761611	1	1	yes	characteristic
142	C₂×C₈	—	2	2	no	—
143	C₂×C₈	—	2	2	no	—
144	C₂×C₈	—	67	1	no	—
145	C₂×C₈	—	67	1	no	—
146	C₂×C₈	—	67	1	no	—
147	C₂×C₈	—	67	1	no	—
148	C₂×C₈	—	134	2	no	—
149	C₂×C₈	—	134	2	no	—
150	C₂×C₈	—	134	4	no	—
151	C₂×C₈	—	134	1	no	—
152	C₂×C₈	—	134	1	no	—
153	C₂×C₈	Dic₆₇⋊C₄	1	1	yes	characteristic
154	C₂×C₈	Dic₆₇⋊C₄	1	1	yes	characteristic
155	C₂×C₈	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
156	C₂×C₈	C₂×Dic₁₃₄	1	1	yes	characteristic
157	C₄²	Dic₆₇⋊C₄	1	1	yes	characteristic
158	C₄²	C₈×D₆₇	1	1	yes	characteristic
159	C₄²	#11761611	1	1	yes	characteristic
160	C₄²	C₂×C₄×D₆₇	1	1	yes	characteristic
161	C₁₆	—	4	2	no	—
162	C₁₆	—	268	2	no	—
163	C₂³	Dic₆₇⋊C₈	1	1	yes	characteristic
164	C₂×C₄	—	2	2	no	—
165	C₂×C₄	—	2	1	no	—
166	C₂×C₄	—	2	1	no	—
167	C₂×C₄	—	2	1	no	—
168	C₂×C₄	—	2	1	no	—
169	C₂×C₄	Dic₆₇⋊C₈	1	1	yes	characteristic
170	C₂×C₄	Dic₆₇⋊C₈	1	1	yes	characteristic
171	C₂×C₄	Dic₆₇⋊C₈	1	1	yes	characteristic
172	C₂×C₄	C₂×Dic₆₇⋊C₄	1	1	yes	characteristic
173	C₂×C₄	C₂×C₈×D₆₇	1	1	yes	characteristic
174	C₂×C₄	C₂×#11761611	1	1	yes	characteristic
175	C₈	—	2	1	no	—
176	C₈	—	2	1	no	—
177	C₈	—	67	2	no	—
178	C₈	—	67	2	no	—
179	C₈	—	134	2	no	—
180	C₈	—	134	2	no	—
181	C₈	—	134	1	no	—
182	C₈	—	134	1	no	—
183	C₈	#14523978	1	2	yes	—
184	C₈	#14524063	1	2	yes	—
185	C₂²	—	2	1	no	—
186	C₂²	—	2	1	no	—
187	C₂²	#16863216	1	2	yes	—
188	C₂²	C₂×Dic₆₇⋊C₈	1	1	yes	characteristic
189	C₄	—	2	1	no	—
190	C₄	—	2	1	no	—
191	C₄	—	2	1	no	—
192	C₄	—	2	1	no	—
193	C₄	#16863432	1	1	yes	characteristic
194	C₄	C₂×#14523978	1	1	yes	characteristic
195	C₄	M₃₂×D₆₇	1	1	yes	characteristic
196	C₄	M₃₂×D₆₇	1	1	yes	characteristic
197	C₂	—	2	2	no	—
198	C₂	#22558955	1	1	yes	characteristic
199	C₂	#22561964	1	1	yes	characteristic
200	C₂	C₂×#16863216	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₈	C₂×C₂²⋊D₆₇
commutator	C₂×C₄×C₆₇	C₂²×C₈
Frattini	C₄×C₈	C₂²×D₆₇
Fitting	C₆₇×#2163	C₂
socle	C₂²×C₆₇	C₂×C₄⋊C₈
prime radical	C₂³×C₆₇	C₄⋊C₈
cosocle	C₄×C₈×C₆₇	C₂³
hypercenter	#2163	D₆₇
hypocenter	C₆₇	#12013
LCS	G ▷ C₂×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂×C₈ ◁ C₂×C₄×C₈ ◁ #2163

Complements

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

N	H ≅ G/N	type
3. #22558278	C₂	semidirect
4. #22558272	C₂	semidirect
106. C₆₇	#12013	semidirect

Maximal subgroups

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

subgroup	index
#22562012	2
C₂×C₈×C₆₇⋊C₈	2
#22558278	2
#22558272	2
C₆₇×#2163	2
#12013	67