Subgroups — #30754369

Order 17152.

398 conjugacy classes of subgroups.

The subgroup lattice has 230 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22560928	C₂	1	2	yes	maximal
2	C₂×#16863147	C₂	1	1	yes	maximal, characteristic
3	C₂×C₈×C₆₇⋊C₈	C₂	1	1	yes	maximal, characteristic
4	#22558050	C₂	1	2	yes	maximal
5	C₄×#106×C₆₇	C₂	1	1	yes	maximal, characteristic
6	C₂×C₄⋊(C₆₇⋊C₈)	—	2	2	no	—
7	C₂×C₄⋊(C₆₇⋊C₈)	C₂²	1	2	yes	—
8	C₂×C₄×C₆₇⋊C₈	C₂²	1	1	yes	characteristic
9	C₂×C₄×C₆₇⋊C₈	C₂²	1	1	yes	characteristic
10	#16863151	—	2	2	no	—
11	#16863147	—	2	2	no	—
12	C₈×C₆₇⋊C₈	—	2	2	no	—
13	C₂×#106×C₆₇	C₄	1	1	yes	characteristic
14	C₂×#106×C₆₇	C₄	1	1	yes	characteristic
15	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
16	C₄×C₄⋊C₄×C₆₇	C₂²	1	2	yes	—
17	#340×C₆₇	C₄	1	2	yes	—
18	C₂²×C₆₇⋊C₈	—	2	2	no	—
19	C₂²×C₆₇⋊C₈	—	2	1	no	—
20	C₂²×C₆₇⋊C₈	D₄	1	2	yes	—
21	C₄⋊(C₆₇⋊C₈)	—	2	2	no	—
22	C₄⋊(C₆₇⋊C₈)	—	2	2	no	—
23	C₄⋊(C₆₇⋊C₈)	—	4	4	no	—
24	C₄×C₆₇⋊C₈	—	2	1	no	—
25	C₄×C₆₇⋊C₈	—	2	1	no	—
26	C₄×C₆₇⋊C₈	—	2	1	no	—
27	C₄×C₆₇⋊C₈	Q₈	1	2	yes	—
28	C₂²×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
29	C₂²×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
30	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	2	yes	—
31	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	2	yes	—
32	C₂×C₄²×C₆₇	—	2	2	no	—
33	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
34	#106×C₆₇	—	2	2	no	—
35	#106×C₆₇	C₈	1	4	yes	complemented
36	C₄×C₈×C₆₇	D₄	1	2	yes	—
37	C₄×C₈×C₆₇	Q₈	1	2	yes	—
38	#95×C₆₇	—	2	2	no	—
39	C₂×C₆₇⋊C₈	—	2	4	no	—
40	C₂×C₆₇⋊C₈	—	2	2	no	—
41	C₂×C₆₇⋊C₈	—	2	2	no	—
42	C₂×C₆₇⋊C₈	—	2	2	no	—
43	C₂×C₆₇⋊C₈	—	4	2	no	—
44	C₂×C₆₇⋊C₈	—	4	2	no	—
45	C₂×C₆₇⋊C₈	—	4	2	no	—
46	C₂×C₆₇⋊C₈	SD₁₆	1	4	yes	—
47	C₂²×C₄×C₆₇	—	2	2	no	—
48	C₂²×C₄×C₆₇	—	2	2	no	—
49	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
50	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
51	C₂²×C₄×C₆₇	#41	1	1	yes	characteristic
52	C₂×C₈×C₆₇	—	2	1	no	—
53	C₂×C₈×C₆₇	—	2	1	no	—
54	C₂×C₈×C₆₇	C₄⋊C₄	1	2	yes	—
55	C₂×C₈×C₆₇	C₄⋊C₄	1	2	yes	—
56	C₂×C₈×C₆₇	C₂×C₈	1	2	yes	—
57	C₂×C₈×C₆₇	M₁₆	1	2	yes	—
58	C₄⋊C₄×C₆₇	—	2	2	no	—
59	C₄⋊C₄×C₆₇	—	2	2	no	—
60	C₄⋊C₄×C₆₇	—	2	2	no	—
61	C₄⋊C₄×C₆₇	C₂×C₈	1	4	yes	—
62	C₄²×C₆₇	—	4	2	no	—
63	C₄²×C₆₇	—	4	2	no	—
64	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
65	C₄²×C₆₇	C₂×Q₈	1	1	yes	characteristic
66	C₄²×C₆₇	C₂×Q₈	1	1	yes	characteristic
67	C₄²×C₆₇	#41	1	1	yes	characteristic
68	C₂³×C₆₇	C₄×D₄	1	1	yes	characteristic
69	C₂×C₄×C₆₇	—	2	2	no	—
70	C₂×C₄×C₆₇	—	2	4	no	—
71	C₂×C₄×C₆₇	—	4	2	no	—
72	C₂×C₄×C₆₇	—	4	2	no	—
73	C₂×C₄×C₆₇	—	4	2	no	—
74	C₂×C₄×C₆₇	—	4	2	no	—
75	C₂×C₄×C₆₇	—	4	2	no	—
76	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
77	C₂×C₄×C₆₇	C₄×Q₈	1	1	yes	characteristic
78	C₂×C₄×C₆₇	#121	1	1	yes	characteristic
79	C₂×C₄×C₆₇	C₂²⋊Q₈	1	1	yes	characteristic
80	C₂×C₄×C₆₇	#125	1	1	yes	characteristic
81	C₂×C₄×C₆₇	C₄⋊Q₈	1	1	yes	characteristic
82	C₂×C₄×C₆₇	C₂²×C₈	1	1	yes	characteristic
83	C₂×C₄×C₆₇	C₂×M₁₆	1	1	yes	characteristic
84	C₂×C₄×C₆₇	#131	1	1	yes	characteristic
85	C₂×C₄×C₆₇	#131	1	1	yes	characteristic
86	C₂×C₄×C₆₇	C₂×SD₁₆	1	2	yes	—
87	C₂×C₄×C₆₇	#135	1	2	yes	—
88	C₈×C₆₇	—	2	2	no	—
89	C₈×C₆₇	C₄⋊C₈	1	4	yes	complemented
90	C₆₇⋊C₈	—	2	4	no	—
91	C₆₇⋊C₈	—	2	4	no	—
92	C₆₇⋊C₈	—	8	2	no	—
93	C₂²×C₆₇	#389	1	1	yes	characteristic
94	C₂²×C₆₇	C₈×D₄	1	1	yes	characteristic
95	C₂²×C₆₇	#436	1	1	yes	characteristic
96	C₂²×C₆₇	C₄×SD₁₆	1	2	yes	—
97	C₂²×C₆₇	#443	1	2	yes	—
98	C₄×C₆₇	—	2	1	no	—
99	C₄×C₆₇	—	2	1	no	—
100	C₄×C₆₇	—	4	4	no	—
101	C₄×C₆₇	—	8	2	no	—
102	C₄×C₆₇	C₂×C₄⋊C₈	1	1	yes	characteristic
103	C₄×C₆₇	#424	1	1	yes	characteristic
104	C₄×C₆₇	C₈×Q₈	1	1	yes	characteristic
105	C₄×C₆₇	#446	1	1	yes	characteristic
106	C₄×C₆₇	#465	1	2	yes	—
107	C₄×C₆₇	#498	1	1	yes	characteristic
108	C₄×C₆₇	#499	1	1	yes	characteristic
109	#9593	—	67	1	no	maximal, Sylow
110	C₂×C₆₇	C₈×SD₁₆	1	2	yes	—
111	C₂×C₆₇	#1590	1	2	yes	—
112	C₂×C₆₇	#1939	1	1	yes	characteristic
113	C₂×C₆₇	#1941	1	1	yes	characteristic
114	C₂×C₆₇	#1945	1	1	yes	characteristic
115	#1939	—	67	2	no	—
116	C₄×#106	D₆₇	1	1	yes	characteristic
117	C₂×#334	—	67	1	no	—
118	C₂×C₈²	—	67	1	no	—
119	#1279	—	67	2	no	—
120	C₆₇	#9593	1	1	yes	characteristic, complemented, Sylow
121	C₂×#106	Dic₆₇	1	1	yes	characteristic
122	C₂×#106	Dic₆₇	1	1	yes	characteristic
123	C₂×C₄⋊C₈	—	67	2	no	—
124	C₂×C₄⋊C₈	—	134	2	no	—
125	C₂×C₄×C₈	—	67	1	no	—
126	C₂×C₄×C₈	—	67	1	no	—
127	C₂×C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
128	C₄×C₄⋊C₄	C₂×D₆₇	1	2	yes	—
129	#340	Dic₆₇	1	2	yes	—
130	#336	—	134	2	no	—
131	#334	—	134	2	no	—
132	C₈²	—	134	2	no	—
133	C₂²×C₈	—	67	2	no	—
134	C₂²×C₈	—	134	2	no	—
135	C₂²×C₈	—	134	1	no	—
136	C₂²×C₈	C₂×Dic₆₇	1	1	yes	characteristic
137	C₂²×C₈	C₂×Dic₆₇	1	1	yes	characteristic
138	C₂×C₄⋊C₄	C₂×Dic₆₇	1	2	yes	—
139	C₂×C₄⋊C₄	C₂×Dic₆₇	1	2	yes	—
140	C₂×C₄²	—	2	2	no	—
141	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
142	#106	—	2	2	no	—
143	#106	C₆₇⋊C₈	1	4	yes	complemented
144	C₄⋊C₈	—	268	4	no	—
145	C₄⋊C₈	—	134	2	no	—
146	C₄⋊C₈	—	134	2	no	—
147	C₄×C₈	—	67	2	no	—
148	C₄×C₈	—	134	1	no	—
149	C₄×C₈	—	134	1	no	—
150	C₄×C₈	—	134	1	no	—
151	C₄×C₈	Dic₁₃₄	1	2	yes	—
152	C₄×C₈	D₂₆₈	1	2	yes	—
153	#95	—	2	2	no	—
154	C₂²×C₄	—	2	2	no	—
155	C₂²×C₄	—	2	2	no	—
156	C₂²×C₄	C₂²×Dic₆₇	1	1	yes	characteristic
157	C₂²×C₄	D₄×D₆₇	1	1	yes	characteristic
158	C₂²×C₄	#11761627	1	1	yes	characteristic
159	C₂×C₈	—	2	1	no	—
160	C₂×C₈	—	2	1	no	—
161	C₂×C₈	—	268	2	no	—
162	C₂×C₈	—	268	2	no	—
163	C₂×C₈	—	268	2	no	—
164	C₂×C₈	—	67	4	no	—
165	C₂×C₈	—	134	4	no	—
166	C₂×C₈	—	134	2	no	—
167	C₂×C₈	—	134	2	no	—
168	C₂×C₈	—	134	2	no	—
169	C₂×C₈	C₄⋊Dic₆₇	1	2	yes	—
170	C₂×C₈	C₄⋊Dic₆₇	1	2	yes	—
171	C₂×C₈	C₂×C₆₇⋊C₈	1	2	yes	—
172	C₂×C₈	#11761612	1	2	yes	—
173	C₄⋊C₄	—	2	2	no	—
174	C₄⋊C₄	—	2	2	no	—
175	C₄⋊C₄	—	2	2	no	—
176	C₄⋊C₄	C₂×C₆₇⋊C₈	1	4	yes	—
177	C₄²	—	4	2	no	—
178	C₄²	—	4	2	no	—
179	C₄²	C₂×D₂₆₈	1	1	yes	characteristic
180	C₄²	Q₈×D₆₇	1	1	yes	characteristic
181	C₄²	C₂×Dic₁₃₄	1	1	yes	characteristic
182	C₄²	#11761629	1	1	yes	characteristic
183	C₂³	D₄×Dic₆₇	1	1	yes	characteristic
184	C₂×C₄	—	2	2	no	—
185	C₂×C₄	—	2	4	no	—
186	C₂×C₄	—	4	2	no	—
187	C₂×C₄	—	4	2	no	—
188	C₂×C₄	—	4	2	no	—
189	C₂×C₄	—	4	2	no	—
190	C₂×C₄	—	4	2	no	—
191	C₂×C₄	C₂×C₄⋊Dic₆₇	1	1	yes	characteristic
192	C₂×C₄	Q₈×Dic₆₇	1	1	yes	characteristic
193	C₂×C₄	#14524018	1	1	yes	characteristic
194	C₂×C₄	#14524023	1	1	yes	characteristic
195	C₂×C₄	#14524034	1	1	yes	characteristic
196	C₂×C₄	#14524042	1	1	yes	characteristic
197	C₂×C₄	C₂²×C₆₇⋊C₈	1	1	yes	characteristic
198	C₂×C₄	C₂×#11761612	1	1	yes	characteristic
199	C₂×C₄	#14524050	1	1	yes	characteristic
200	C₂×C₄	#14524050	1	1	yes	characteristic
201	C₂×C₄	SD₁₆×D₆₇	1	2	yes	—
202	C₂×C₄	#14524064	1	2	yes	—
203	C₈	—	2	2	no	—
204	C₈	—	536	2	no	—
205	C₈	—	134	4	no	—
206	C₈	—	134	4	no	—
207	C₈	C₄⋊(C₆₇⋊C₈)	1	4	yes	complemented
208	C₂²	#16863295	1	1	yes	characteristic
209	C₂²	D₄×C₆₇⋊C₈	1	1	yes	characteristic
210	C₂²	#16863474	1	1	yes	characteristic
211	C₂²	SD₁₆×Dic₆₇	1	2	yes	—
212	C₂²	#16863513	1	2	yes	—
213	C₄	—	2	1	no	—
214	C₄	—	2	1	no	—
215	C₄	—	4	4	no	—
216	C₄	—	8	2	no	—
217	C₄	C₂×C₄⋊(C₆₇⋊C₈)	1	1	yes	characteristic
218	C₄	#16863423	1	1	yes	characteristic
219	C₄	Q₈×C₆₇⋊C₈	1	1	yes	characteristic
220	C₄	#16863527	1	1	yes	characteristic
221	C₄	#16863626	1	2	yes	—
222	C₄	#16863793	1	1	yes	characteristic
223	C₄	#16863796	1	1	yes	characteristic
224	C₂	SD₁₆×C₆₇⋊C₈	1	2	yes	—
225	C₂	#22559100	1	2	yes	—
226	C₂	#22560928	1	1	yes	characteristic
227	C₂	#22560938	1	1	yes	characteristic
228	C₂	#22560958	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	D₄×D₆₇
commutator	C₂×C₄×C₆₇	C₂²×C₈
Frattini	C₂×C₄²	C₂²×D₆₇
Fitting	C₄×#106×C₆₇	C₂
socle, prime radical	C₂³×C₆₇	C₄×D₄
cosocle	C₂×C₄²×C₆₇	C₂³
hypercenter	C₄×#106	D₆₇
hypocenter	C₆₇	#9593
LCS	G ▷ C₂×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄² ◁ C₄×#106

Complements

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

N	H ≅ G/N	type
35. #106×C₆₇	C₈	semidirect
89. C₈×C₆₇	C₄⋊C₈	semidirect
120. C₆₇	#9593	semidirect
143. #106	C₆₇⋊C₈	semidirect
207. C₈	C₄⋊(C₆₇⋊C₈)	semidirect

Maximal subgroups

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

subgroup	index
#22560928	2
C₂×#16863147	2
C₂×C₈×C₆₇⋊C₈	2
#22558050	2
C₄×#106×C₆₇	2
#9593	67