Subgroups — #30754443

Order 17152.

306 conjugacy classes of subgroups.

The subgroup lattice has 216 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22558961	C₂	1	2	yes	maximal
2	C₂×#16863147	C₂	1	1	yes	maximal, characteristic
3	#22558492	C₂	1	1	yes	maximal, characteristic
4	#22558060	C₂	1	2	yes	maximal
5	C₄×#108×C₆₇	C₂	1	1	yes	maximal, characteristic
6	#16863449	C₂²	1	1	yes	characteristic
7	#16863449	C₂²	1	1	yes	characteristic
8	C₂×C₄⋊(C₆₇⋊C₈)	C₂²	1	2	yes	—
9	#16863149	—	2	2	no	—
10	#16863147	—	2	1	no	—
11	#16863147	—	2	1	no	—
12	#16863147	—	2	2	no	—
13	#16863117	—	2	1	no	—
14	C₈×C₆₇⋊C₈	—	2	1	no	—
15	C₂×#108×C₆₇	C₄	1	1	yes	characteristic
16	C₂×#108×C₆₇	C₄	1	1	yes	characteristic
17	C₄×M₁₆×C₆₇	C₂²	1	2	yes	—
18	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
19	#341×C₆₇	C₄	1	2	yes	—
20	C₂²×C₆₇⋊C₈	—	2	2	no	—
21	C₄⋊(C₆₇⋊C₈)	—	2	2	no	—
22	C₄⋊(C₆₇⋊C₈)	—	2	2	no	—
23	C₄⋊(C₆₇⋊C₈)	—	2	1	no	—
24	C₄⋊(C₆₇⋊C₈)	D₄	1	2	yes	—
25	C₂²⋊(C₆₇⋊C₈)	—	2	1	no	—
26	C₂²⋊(C₆₇⋊C₈)	Q₈	1	2	yes	—
27	C₄×C₆₇⋊C₈	—	2	1	no	—
28	C₄×C₆₇⋊C₈	—	2	1	no	—
29	C₂×M₁₆×C₆₇	C₂×C₄	1	2	yes	—
30	C₂×M₁₆×C₆₇	C₂×C₄	1	2	yes	—
31	C₂²×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
32	C₂²×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
33	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
34	#108×C₆₇	—	2	2	no	—
35	#108×C₆₇	C₈	1	4	yes	complemented
36	#97×C₆₇	—	2	2	no	—
37	C₄×C₈×C₆₇	—	2	2	no	—
38	C₄×C₈×C₆₇	D₄	1	1	yes	characteristic
39	C₄×C₈×C₆₇	D₄	1	1	yes	characteristic
40	C₄×C₈×C₆₇	Q₈	1	1	yes	characteristic
41	C₄×C₈×C₆₇	Q₈	1	1	yes	characteristic
42	C₂×C₆₇⋊C₈	—	2	2	no	—
43	C₂×C₆₇⋊C₈	—	2	2	no	—
44	C₂×C₆₇⋊C₈	—	4	2	no	—
45	C₂×C₆₇⋊C₈	—	8	2	no	—
46	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
47	C₂²×C₄×C₆₇	C₂×Q₈	1	1	yes	characteristic
48	C₂²×C₄×C₆₇	#41	1	1	yes	characteristic
49	M₁₆×C₆₇	—	2	2	no	—
50	M₁₆×C₆₇	—	2	2	no	—
51	M₁₆×C₆₇	—	2	2	no	—
52	M₁₆×C₆₇	C₂×C₈	1	4	yes	—
53	C₂×C₈×C₆₇	—	2	2	no	—
54	C₂×C₈×C₆₇	—	2	1	no	—
55	C₂×C₈×C₆₇	—	2	1	no	—
56	C₂×C₈×C₆₇	—	2	2	no	—
57	C₂×C₈×C₆₇	C₄⋊C₄	1	1	yes	characteristic
58	C₂×C₈×C₆₇	C₄⋊C₄	1	1	yes	characteristic
59	C₂×C₈×C₆₇	C₄⋊C₄	1	1	yes	characteristic
60	C₂×C₈×C₆₇	C₄⋊C₄	1	1	yes	characteristic
61	C₂×C₈×C₆₇	C₂×C₈	1	2	yes	—
62	C₂×C₈×C₆₇	M₁₆	1	2	yes	—
63	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
64	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
65	C₄²×C₆₇	C₂×Q₈	1	1	yes	characteristic
66	C₄²×C₆₇	#41	1	1	yes	characteristic
67	C₂³×C₆₇	C₄×Q₈	1	1	yes	characteristic
68	C₂×C₄×C₆₇	—	2	2	no	—
69	C₂×C₄×C₆₇	—	4	1	no	—
70	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
71	C₂×C₄×C₆₇	C₄×D₄	1	1	yes	characteristic
72	C₂×C₄×C₆₇	#121	1	1	yes	characteristic
73	C₂×C₄×C₆₇	C₂²⋊Q₈	1	1	yes	characteristic
74	C₂×C₄×C₆₇	#125	1	1	yes	characteristic
75	C₂×C₄×C₆₇	C₄⋊Q₈	1	1	yes	characteristic
76	C₂×C₄×C₆₇	C₂²×C₈	1	1	yes	characteristic
77	C₂×C₄×C₆₇	C₂×M₁₆	1	1	yes	characteristic
78	C₂×C₄×C₆₇	#131	1	1	yes	characteristic
79	C₂×C₄×C₆₇	#131	1	1	yes	characteristic
80	C₈×C₆₇	—	2	1	no	—
81	C₈×C₆₇	—	2	1	no	—
82	C₈×C₆₇	—	2	4	no	—
83	C₈×C₆₇	—	4	2	no	—
84	C₈×C₆₇	C₄⋊C₈	1	2	yes	—
85	C₈×C₆₇	C₄⋊C₈	1	2	yes	—
86	C₆₇⋊C₈	—	4	2	no	—
87	C₆₇⋊C₈	—	4	2	no	—
88	C₆₇⋊C₈	—	8	2	no	—
89	C₂²×C₆₇	—	2	1	no	—
90	C₂²×C₆₇	—	2	1	no	—
91	C₂²×C₆₇	#389	1	1	yes	characteristic
92	C₂²×C₆₇	C₈×Q₈	1	1	yes	characteristic
93	C₂²×C₆₇	#446	1	1	yes	characteristic
94	C₄×C₆₇	—	2	1	no	—
95	C₄×C₆₇	—	2	1	no	—
96	C₄×C₆₇	—	2	1	no	—
97	C₄×C₆₇	C₂×C₄⋊C₈	1	1	yes	characteristic
98	C₄×C₆₇	#424	1	1	yes	characteristic
99	C₄×C₆₇	C₈×D₄	1	1	yes	characteristic
100	C₄×C₆₇	#436	1	1	yes	characteristic
101	C₄×C₆₇	#471	1	2	yes	—
102	#9607	—	67	1	no	maximal, Sylow
103	C₂×C₆₇	—	2	1	no	—
104	C₂×C₆₇	—	2	1	no	—
105	C₂×C₆₇	#1939	1	1	yes	characteristic
106	C₂×C₆₇	#1951	1	1	yes	characteristic
107	C₂×C₆₇	#1953	1	1	yes	characteristic
108	C₄×#108	D₆₇	1	1	yes	characteristic
109	#1565	—	67	2	no	—
110	C₂×#334	—	67	1	no	—
111	#1450	—	67	1	no	—
112	#1283	—	67	2	no	—
113	C₆₇	#9607	1	1	yes	characteristic, complemented, Sylow
114	#431	—	67	1	no	—
115	#431	—	67	1	no	—
116	C₂×#108	Dic₆₇	1	1	yes	characteristic
117	C₂×#108	Dic₆₇	1	1	yes	characteristic
118	C₂×C₄⋊C₈	—	67	2	no	—
119	C₄×M₁₆	C₂×D₆₇	1	2	yes	—
120	C₂×C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
121	#341	Dic₆₇	1	2	yes	—
122	#335	—	134	2	no	—
123	#334	—	134	2	no	—
124	#334	—	134	1	no	—
125	#334	—	134	1	no	—
126	#322	—	134	1	no	—
127	C₈²	—	134	1	no	—
128	C₂×M₁₆	C₂×Dic₆₇	1	2	yes	—
129	C₂×M₁₆	C₂×Dic₆₇	1	2	yes	—
130	C₂²×C₈	—	134	2	no	—
131	C₂²×C₈	C₂×Dic₆₇	1	1	yes	characteristic
132	C₂²×C₈	C₂×Dic₆₇	1	1	yes	characteristic
133	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
134	#108	—	2	2	no	—
135	#108	C₆₇⋊C₈	1	4	yes	complemented
136	C₄⋊C₈	—	67	2	no	—
137	C₄⋊C₈	—	134	2	no	—
138	C₄⋊C₈	—	134	2	no	—
139	C₄⋊C₈	—	134	1	no	—
140	C₂²⋊C₈	—	67	2	no	—
141	C₂²⋊C₈	—	134	1	no	—
142	#97	—	2	2	no	—
143	C₄×C₈	—	2	2	no	—
144	C₄×C₈	—	134	1	no	—
145	C₄×C₈	—	134	1	no	—
146	C₄×C₈	Dic₁₃₄	1	1	yes	characteristic
147	C₄×C₈	Dic₁₃₄	1	1	yes	characteristic
148	C₄×C₈	D₂₆₈	1	1	yes	characteristic
149	C₄×C₈	D₂₆₈	1	1	yes	characteristic
150	C₂²×C₄	C₂²×Dic₆₇	1	1	yes	characteristic
151	C₂²×C₄	Q₈×D₆₇	1	1	yes	characteristic
152	C₂²×C₄	#11761629	1	1	yes	characteristic
153	M₁₆	—	2	2	no	—
154	M₁₆	—	2	2	no	—
155	M₁₆	—	2	2	no	—
156	M₁₆	C₂×C₆₇⋊C₈	1	4	yes	—
157	C₂×C₈	—	2	2	no	—
158	C₂×C₈	—	2	1	no	—
159	C₂×C₈	—	2	1	no	—
160	C₂×C₈	—	2	2	no	—
161	C₂×C₈	—	268	2	no	—
162	C₂×C₈	—	536	2	no	—
163	C₂×C₈	—	134	2	no	—
164	C₂×C₈	—	134	2	no	—
165	C₂×C₈	C₄⋊Dic₆₇	1	1	yes	characteristic
166	C₂×C₈	C₄⋊Dic₆₇	1	1	yes	characteristic
167	C₂×C₈	C₄⋊Dic₆₇	1	1	yes	characteristic
168	C₂×C₈	C₄⋊Dic₆₇	1	1	yes	characteristic
169	C₂×C₈	C₂×C₆₇⋊C₈	1	2	yes	—
170	C₂×C₈	#11761612	1	2	yes	—
171	C₄²	D₄×D₆₇	1	1	yes	characteristic
172	C₄²	C₂×D₂₆₈	1	1	yes	characteristic
173	C₄²	C₂×Dic₁₃₄	1	1	yes	characteristic
174	C₄²	#11761627	1	1	yes	characteristic
175	C₂³	Q₈×Dic₆₇	1	1	yes	characteristic
176	C₂×C₄	—	2	2	no	—
177	C₂×C₄	—	4	1	no	—
178	C₂×C₄	C₂×C₄⋊Dic₆₇	1	1	yes	characteristic
179	C₂×C₄	D₄×Dic₆₇	1	1	yes	characteristic
180	C₂×C₄	#14524018	1	1	yes	characteristic
181	C₂×C₄	#14524023	1	1	yes	characteristic
182	C₂×C₄	#14524034	1	1	yes	characteristic
183	C₂×C₄	#14524042	1	1	yes	characteristic
184	C₂×C₄	C₂²×C₆₇⋊C₈	1	1	yes	characteristic
185	C₂×C₄	C₂×#11761612	1	1	yes	characteristic
186	C₂×C₄	#14524050	1	1	yes	characteristic
187	C₂×C₄	#14524050	1	1	yes	characteristic
188	C₈	—	2	1	no	—
189	C₈	—	2	1	no	—
190	C₈	—	2	4	no	—
191	C₈	—	4	2	no	—
192	C₈	—	268	2	no	—
193	C₈	—	268	2	no	—
194	C₈	—	536	2	no	—
195	C₈	C₄⋊(C₆₇⋊C₈)	1	2	yes	—
196	C₈	C₄⋊(C₆₇⋊C₈)	1	2	yes	—
197	C₂²	—	2	1	no	—
198	C₂²	—	2	1	no	—
199	C₂²	#16863295	1	1	yes	characteristic
200	C₂²	Q₈×C₆₇⋊C₈	1	1	yes	characteristic
201	C₂²	#16863527	1	1	yes	characteristic
202	C₄	—	2	1	no	—
203	C₄	—	2	1	no	—
204	C₄	—	2	1	no	—
205	C₄	C₂×C₄⋊(C₆₇⋊C₈)	1	1	yes	characteristic
206	C₄	#16863423	1	1	yes	characteristic
207	C₄	D₄×C₆₇⋊C₈	1	1	yes	characteristic
208	C₄	#16863474	1	1	yes	characteristic
209	C₄	#16863660	1	2	yes	—
210	C₂	—	2	1	no	—
211	C₂	—	2	1	no	—
212	C₂	#22560928	1	1	yes	characteristic
213	C₂	#22560990	1	1	yes	characteristic
214	C₂	#22561000	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	#14524018
commutator	C₂×C₄×C₆₇	C₂²×C₈
Frattini	C₂×C₄²	C₂²×D₆₇
Fitting	C₄×#108×C₆₇	C₂
socle	C₂²×C₆₇	#389
prime radical	C₂³×C₆₇	C₄×Q₈
cosocle	C₂×C₄²×C₆₇	C₂³
hypercenter	C₄×#108	D₆₇
hypocenter	C₆₇	#9607
LCS	G ▷ C₂×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₄×#108

Complements

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

N	H ≅ G/N	type
35. #108×C₆₇	C₈	semidirect
113. C₆₇	#9607	semidirect
135. #108	C₆₇⋊C₈	semidirect

Maximal subgroups

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

subgroup	index
#22558961	2
C₂×#16863147	2
#22558492	2
#22558060	2
C₄×#108×C₆₇	2
#9607	67