Subgroups — #30767441

Order 17152.

246 conjugacy classes of subgroups.

The subgroup lattice has 222 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22562007	C₂	1	1	yes	maximal, characteristic
2	C₄⋊C₁₆×D₆₇	C₂	1	1	yes	maximal, characteristic
3	C₈×#11761611	C₂	1	1	yes	maximal, characteristic
4	#22558279	C₂	1	1	yes	maximal, characteristic, complemented
5	#22558278	C₂	1	1	yes	maximal, characteristic, complemented
6	#22558272	C₂	1	1	yes	maximal, characteristic, complemented
7	C₆₇×#1367	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₁₆×D₆₇	—	2	1	no	—
9	#16863447	C₄	1	1	yes	characteristic
10	C₄×#11761611	C₄	1	1	yes	characteristic
11	C₄×C₈×D₆₇	C₂²	1	1	yes	characteristic
12	C₄⋊(C₆₇⋊C₁₆)	C₂²	1	1	yes	characteristic
13	C₄⋊(C₆₇⋊C₁₆)	C₂²	1	1	yes	characteristic
14	Dic₆₇⋊C₁₆	—	2	1	no	—
15	(C₂×D₆₇)⋊C₁₆	—	2	1	no	—
16	C₁₆×Dic₆₇	—	2	1	no	—
17	#16863118	C₄	1	1	yes	characteristic
18	#16863117	C₄	1	1	yes	characteristic
19	C₈×C₆₇⋊C₈	C₂²	1	1	yes	characteristic
20	C₄⋊C₁₆×C₆₇	C₂²	1	1	yes	characteristic
21	C₄⋊C₁₆×C₆₇	C₂²	1	1	yes	characteristic
22	C₈²×C₆₇	C₂²	1	1	yes	characteristic
23	C₂×#11761611	C₈	1	1	yes	characteristic
24	C₂×#11761611	C₈	1	1	yes	characteristic
25	C₂×C₈×D₆₇	D₄	1	1	yes	characteristic
26	C₂×C₈×D₆₇	Q₈	1	1	yes	characteristic
27	C₄²×D₆₇	C₂×C₄	1	1	yes	characteristic
28	C₂×C₆₇⋊C₁₆	—	2	1	no	—
29	C₂×C₆₇⋊C₁₆	—	2	1	no	—
30	C₁₆×D₆₇	—	4	2	no	—
31	Dic₆₇⋊C₈	—	2	1	no	—
32	(C₂×D₆₇)⋊C₈	—	2	1	no	—
33	#14523953	C₈	1	1	yes	characteristic
34	#14523953	C₈	1	1	yes	characteristic
35	C₄×C₆₇⋊C₈	C₂×C₄	1	1	yes	characteristic
36	C₄×C₆₇⋊C₈	C₂×C₄	1	1	yes	characteristic
37	C₄×C₆₇⋊C₈	C₂×C₄	1	1	yes	characteristic
38	C₈×Dic₆₇	D₄	1	1	yes	characteristic
39	C₈×Dic₆₇	Q₈	1	1	yes	characteristic
40	C₂×C₁₆×C₆₇	—	2	1	no	—
41	C₂×C₁₆×C₆₇	—	2	1	no	—
42	C₄×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
43	C₄×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
44	C₄×C₈×C₆₇	C₂³	1	1	yes	characteristic
45	C₂×C₄×D₆₇	C₄⋊C₄	1	1	yes	characteristic
46	C₂×C₄×D₆₇	C₂×C₈	1	1	yes	characteristic
47	C₂×C₄×D₆₇	M₁₆	1	1	yes	characteristic
48	#11761611	—	2	2	no	—
49	#11761611	C₁₆	1	4	yes	complemented
50	C₂×C₆₇⋊C₈	—	2	1	no	—
51	C₂×C₆₇⋊C₈	C₄⋊C₄	1	1	yes	characteristic
52	C₂×C₆₇⋊C₈	C₄⋊C₄	1	1	yes	characteristic
53	C₂×C₆₇⋊C₈	C₂×C₈	1	1	yes	characteristic
54	C₂×C₆₇⋊C₈	C₂×C₈	1	1	yes	characteristic
55	C₈×D₆₇	—	2	1	no	—
56	C₈×D₆₇	—	2	1	no	—
57	C₈×D₆₇	—	4	1	no	—
58	C₄×Dic₆₇	C₄⋊C₄	1	1	yes	characteristic
59	C₄×Dic₆₇	C₂×C₈	1	1	yes	characteristic
60	C₄×Dic₆₇	M₁₆	1	1	yes	characteristic
61	C₆₇⋊C₁₆	—	4	1	no	—
62	C₆₇⋊C₁₆	—	4	1	no	—
63	C₂×C₈×C₆₇	—	2	1	no	—
64	C₂×C₈×C₆₇	C₂×C₈	1	1	yes	characteristic
65	C₂×C₈×C₆₇	C₂×C₈	1	1	yes	characteristic
66	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
67	C₂×C₈×C₆₇	C₂×Q₈	1	1	yes	characteristic
68	C₄²×C₆₇	C₂²×C₄	1	1	yes	characteristic
69	C₁₆×C₆₇	—	4	1	no	—
70	C₁₆×C₆₇	—	4	1	no	—
71	C₂²×D₆₇	C₄⋊C₈	1	1	yes	characteristic
72	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
73	C₂×C₄×C₆₇	C₂²×C₈	1	1	yes	characteristic
74	C₂×C₄×C₆₇	C₂×M₁₆	1	1	yes	characteristic
75	C₂×Dic₆₇	C₄⋊C₈	1	1	yes	characteristic
76	C₂×Dic₆₇	C₄⋊C₈	1	1	yes	characteristic
77	C₂×Dic₆₇	C₄⋊C₈	1	1	yes	characteristic
78	C₄×D₆₇	—	2	1	no	—
79	C₄×D₆₇	—	2	1	no	—
80	C₄×D₆₇	—	4	1	no	—
81	C₄×D₆₇	C₂×C₁₆	1	2	yes	—
82	C₄×D₆₇	M₃₂	1	2	yes	—
83	C₈×C₆₇	—	2	1	no	—
84	C₈×C₆₇	—	2	1	no	—
85	C₈×C₆₇	—	4	1	no	—
86	C₈×C₆₇	C₂×C₁₆	1	2	yes	complemented
87	C₈×C₆₇	AGL(1,ℤ/8)	1	1	yes	characteristic
88	C₈×C₆₇	#137	1	1	yes	characteristic
89	C₆₇⋊C₈	—	2	1	no	—
90	C₆₇⋊C₈	—	2	1	no	—
91	C₆₇⋊C₈	—	2	1	no	—
92	C₆₇⋊C₈	—	4	1	no	—
93	C₆₇⋊C₈	C₂×C₁₆	1	2	yes	complemented
94	C₂²×C₆₇	C₂×C₄⋊C₈	1	1	yes	characteristic
95	C₂×D₆₇	—	2	1	no	—
96	C₂×D₆₇	—	2	1	no	—
97	C₂×D₆₇	C₄⋊C₁₆	1	2	yes	—
98	C₄×C₆₇	—	2	1	no	—
99	C₄×C₆₇	#428	1	1	yes	characteristic
100	C₄×C₆₇	#430	1	1	yes	characteristic
101	C₄×C₆₇	C₂²×C₁₆	1	1	yes	characteristic
102	C₄×C₆₇	C₂×M₃₂	1	1	yes	characteristic
103	Dic₆₇	—	2	1	no	—
104	Dic₆₇	—	2	1	no	—
105	Dic₆₇	C₄⋊C₁₆	1	2	yes	—
106	#12015	—	67	1	no	maximal, Sylow
107	C₂×C₆₇	#1565	1	1	yes	characteristic
108	C₂×C₆₇	C₂×C₄⋊C₁₆	1	1	yes	characteristic
109	C₂×C₆₇	#2153	1	1	yes	characteristic
110	D₆₇	—	2	2	no	—
111	#2162	—	67	1	no	—
112	C₂×C₄⋊C₁₆	—	67	1	no	—
113	C₈×M₁₆	—	67	1	no	—
114	#1370	—	67	1	no	—
115	#1370	—	67	1	no	—
116	#1367	—	67	1	no	—
117	#1367	D₆₇	1	1	yes	characteristic, complemented
118	C₆₇	#12015	1	1	yes	characteristic, complemented, Sylow
119	C₂²×C₁₆	—	134	1	no	—
120	#431	—	67	1	no	—
121	C₄×M₁₆	—	67	1	no	—
122	C₂×C₄×C₈	—	67	1	no	—
123	C₄⋊C₁₆	—	67	1	no	—
124	C₄⋊C₁₆	—	67	1	no	—
125	C₄⋊C₁₆	—	134	1	no	—
126	C₄⋊C₁₆	C₂×D₆₇	1	1	yes	characteristic
127	C₄⋊C₁₆	C₂×D₆₇	1	1	yes	characteristic
128	C₂²⋊C₁₆	—	134	1	no	—
129	C₄×C₁₆	—	134	1	no	—
130	#322	—	67	1	no	—
131	#322	—	67	1	no	—
132	C₈²	—	67	1	no	—
133	C₈²	C₂×D₆₇	1	1	yes	characteristic
134	C₂×M₁₆	—	67	1	no	—
135	C₂×M₁₆	—	67	1	no	—
136	C₂²×C₈	—	67	1	no	—
137	C₂²×C₈	—	67	1	no	—
138	C₂×C₄²	—	67	1	no	—
139	C₂×C₁₆	—	2	1	no	—
140	C₂×C₁₆	—	2	1	no	—
141	C₂×C₁₆	—	268	2	no	—
142	C₂×C₁₆	—	134	1	no	—
143	C₂×C₁₆	—	134	1	no	—
144	C₄⋊C₈	—	134	1	no	—
145	C₂²⋊C₈	—	134	1	no	—
146	#97	—	67	1	no	—
147	#97	—	67	1	no	—
148	C₄×C₈	—	67	1	no	—
149	C₄×C₈	—	67	1	no	—
150	C₄×C₈	—	67	1	no	—
151	C₄×C₈	—	67	1	no	—
152	C₄×C₈	—	67	1	no	—
153	C₄×C₈	C₄×D₆₇	1	1	yes	characteristic
154	C₄×C₈	C₄×D₆₇	1	1	yes	characteristic
155	C₄×C₈	C₂²×D₆₇	1	1	yes	characteristic
156	C₂²×C₄	—	67	1	no	—
157	C₂²×C₄	—	67	1	no	—
158	C₂²×C₄	—	67	1	no	—
159	M₁₆	—	67	4	no	—
160	M₁₆	—	134	2	no	—
161	C₂×C₈	—	2	1	no	—
162	C₂×C₈	—	268	1	no	—
163	C₂×C₈	—	67	1	no	—
164	C₂×C₈	—	67	1	no	—
165	C₂×C₈	—	67	1	no	—
166	C₂×C₈	—	67	1	no	—
167	C₂×C₈	—	134	1	no	—
168	C₂×C₈	—	134	1	no	—
169	C₂×C₈	—	134	1	no	—
170	C₂×C₈	C₈×D₆₇	1	1	yes	characteristic
171	C₂×C₈	C₈×D₆₇	1	1	yes	characteristic
172	C₂×C₈	D₄×D₆₇	1	1	yes	characteristic
173	C₂×C₈	Q₈×D₆₇	1	1	yes	characteristic
174	C₄²	—	67	1	no	—
175	C₄²	—	67	1	no	—
176	C₄²	—	67	1	no	—
177	C₄²	C₂×C₄×D₆₇	1	1	yes	characteristic
178	C₁₆	—	4	1	no	—
179	C₁₆	—	4	1	no	—
180	C₁₆	—	268	1	no	—
181	C₁₆	—	268	1	no	—
182	C₂³	—	67	1	no	—
183	C₂×C₄	—	268	1	no	—
184	C₂×C₄	—	67	1	no	—
185	C₂×C₄	—	67	2	no	—
186	C₂×C₄	—	67	2	no	—
187	C₂×C₄	—	67	1	no	—
188	C₂×C₄	—	67	1	no	—
189	C₂×C₄	—	134	1	no	—
190	C₂×C₄	—	134	1	no	—
191	C₂×C₄	C₄⋊C₄×D₆₇	1	1	yes	characteristic
192	C₂×C₄	C₂×C₈×D₆₇	1	1	yes	characteristic
193	C₂×C₄	M₁₆×D₆₇	1	1	yes	characteristic
194	C₈	—	2	1	no	—
195	C₈	—	2	1	no	—
196	C₈	—	4	1	no	—
197	C₈	—	268	1	no	—
198	C₈	—	67	2	no	—
199	C₈	—	134	1	no	—
200	C₈	—	134	1	no	—
201	C₈	—	134	1	no	—
202	C₈	C₁₆×D₆₇	1	2	yes	complemented
203	C₈	#14524070	1	1	yes	characteristic
204	C₈	#14524072	1	1	yes	characteristic
205	C₂²	—	67	2	no	—
206	C₂²	—	134	1	no	—
207	C₂²	—	134	1	no	—
208	C₂²	C₄⋊C₈×D₆₇	1	1	yes	characteristic
209	C₄	—	2	1	no	—
210	C₄	—	67	2	no	—
211	C₄	—	134	1	no	—
212	C₄	—	134	1	no	—
213	C₄	#16863435	1	1	yes	characteristic
214	C₄	#16863443	1	1	yes	characteristic
215	C₄	C₂×C₁₆×D₆₇	1	1	yes	characteristic
216	C₄	M₃₂×D₆₇	1	1	yes	characteristic
217	C₂	—	134	2	no	—
218	C₂	#22558958	1	1	yes	characteristic
219	C₂	C₄⋊C₁₆×D₆₇	1	1	yes	characteristic
220	C₂	#22561973	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₈	D₄×D₆₇
commutator	C₄×C₆₇	C₂²×C₁₆
Frattini	C₄×C₈	C₂²×D₆₇
Fitting	C₆₇×#1367	C₂
socle	C₂²×C₆₇	C₂×C₄⋊C₈
prime radical	C₂²×D₆₇	C₄⋊C₈
cosocle	C₄×C₈×C₆₇	C₂³
hypercenter	#1367	D₆₇
hypocenter	C₆₇	#12015
LCS	G ▷ C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂×C₈ ◁ C₄×C₈ ◁ #1367

Complements

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

N	H ≅ G/N	type
4. #22558279	C₂	semidirect
5. #22558278	C₂	semidirect
6. #22558272	C₂	semidirect
7. C₆₇×#1367	C₂	semidirect
49. #11761611	C₁₆	semidirect
86. C₈×C₆₇	C₂×C₁₆	semidirect
93. C₆₇⋊C₈	C₂×C₁₆	semidirect
117. #1367	D₆₇	semidirect
118. C₆₇	#12015	semidirect
202. C₈	C₁₆×D₆₇	semidirect

Maximal subgroups

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

subgroup	index
#22562007	2
C₄⋊C₁₆×D₆₇	2
C₈×#11761611	2
#22558279	2
#22558278	2
#22558272	2
C₆₇×#1367	2
#12015	67