Subgroups — #30767116

Order 17152.

306 conjugacy classes of subgroups.

The subgroup lattice has 224 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22570723	C₂	1	1	yes	maximal, characteristic
2	#22558957	C₂	1	1	yes	maximal, characteristic
3	#22558266	C₂	1	2	yes	maximal, complemented
4	#22558263	C₂	1	2	yes	maximal, complemented
5	C₂×C₄⋊C₁₆×C₆₇	C₂	1	1	yes	maximal, characteristic
6	#16863953	—	2	1	no	—
7	#16863801	C₂²	1	1	yes	characteristic
8	#16863801	C₂²	1	1	yes	characteristic
9	#16863798	—	2	1	no	—
10	#16863798	C₄	1	2	yes	—
11	#16863794	—	2	1	no	—
12	#16863449	—	2	1	no	—
13	#16863434	—	2	1	no	—
14	#16863434	C₄	1	2	yes	—
15	#16863149	C₂²	1	2	yes	—
16	#16863147	—	2	1	no	—
17	C₂²×C₁₆×C₆₇	—	2	1	no	—
18	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
19	C₄⋊C₁₆×C₆₇	—	2	1	no	—
20	C₄⋊C₁₆×C₆₇	C₂²	1	2	yes	—
21	#14524043	—	2	1	no	—
22	#14524043	—	2	1	no	—
23	#14524035	—	2	2	no	—
24	#14524022	—	4	1	no	—
25	#14524022	—	4	1	no	—
26	C₄×Dic₁₃₄	—	4	1	no	—
27	C₄×Dic₁₃₄	—	4	1	no	—
28	#14524002	—	2	2	no	—
29	#14523977	—	2	1	no	—
30	#14523977	—	2	1	no	—
31	#14523977	C₂×C₄	1	2	yes	—
32	#14523977	C₂×C₄	1	2	yes	—
33	#14523975	—	2	2	no	—
34	#14523975	—	2	2	no	—
35	C₄⋊(C₆₇⋊C₈)	—	2	2	no	—
36	C₂²⋊(C₆₇⋊C₈)	—	4	1	no	—
37	C₄×C₆₇⋊C₈	—	4	1	no	—
38	C₂²×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
39	C₂²×C₈×C₆₇	D₄	1	1	yes	characteristic
40	C₂×C₄²×C₆₇	D₄	1	1	yes	characteristic
41	C₂×C₁₆×C₆₇	—	2	2	no	—
42	C₂×C₁₆×C₆₇	—	4	2	no	—
43	C₄×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
44	C₄×C₈×C₆₇	D₄	1	1	yes	characteristic
45	C₄×C₈×C₆₇	D₄	1	1	yes	characteristic
46	C₄×C₈×C₆₇	C₂³	1	1	yes	characteristic
47	C₂×Dic₁₃₄	—	4	1	no	—
48	C₂×Dic₁₃₄	—	4	1	no	—
49	C₂×C₆₇⋊C₈	—	4	2	no	—
50	C₄⋊Dic₆₇	—	2	2	no	—
51	C₄⋊Dic₆₇	—	2	2	no	—
52	C₄⋊Dic₆₇	—	4	1	no	—
53	C₄⋊Dic₆₇	—	4	1	no	—
54	Dic₆₇⋊C₄	—	4	2	no	—
55	Dic₆₇⋊C₄	—	4	2	no	—
56	C₂²⋊Dic₆₇	—	4	2	no	—
57	C₄×Dic₆₇	—	4	2	no	—
58	C₂²×C₄×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
59	C₂²×C₄×C₆₇	D₈	1	1	yes	characteristic
60	C₂²×C₄×C₆₇	SD₁₆	1	1	yes	characteristic
61	C₂×C₈×C₆₇	—	2	2	no	—
62	C₂×C₈×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
63	C₂×C₈×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
64	C₂×C₈×C₆₇	SD₁₆	1	2	yes	—
65	C₂×C₈×C₆₇	Q₁₆	1	2	yes	—
66	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
67	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
68	C₄²×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
69	C₄²×C₆₇	D₈	1	1	yes	characteristic
70	C₄²×C₆₇	SD₁₆	1	1	yes	characteristic
71	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
72	C₁₆×C₆₇	—	4	2	no	—
73	C₂³×C₆₇	#102	1	1	yes	characteristic
74	C₂×C₄×C₆₇	—	2	2	no	—
75	C₂×C₄×C₆₇	—	4	1	no	—
76	C₂×C₄×C₆₇	#101	1	2	yes	—
77	C₂×C₄×C₆₇	#102	1	1	yes	characteristic
78	C₂×C₄×C₆₇	#102	1	1	yes	characteristic
79	C₂×C₄×C₆₇	#102	1	1	yes	characteristic
80	C₂×C₄×C₆₇	#103	1	2	yes	—
81	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
82	C₂×C₄×C₆₇	C₂×D₈	1	1	yes	characteristic
83	C₂×C₄×C₆₇	C₂×SD₁₆	1	1	yes	characteristic
84	C₂×Dic₆₇	—	4	2	no	—
85	C₂×Dic₆₇	—	4	2	no	—
86	Dic₁₃₄	—	8	2	no	—
87	Dic₁₃₄	—	8	2	no	—
88	C₈×C₆₇	—	2	1	no	—
89	C₈×C₆₇	—	2	1	no	—
90	C₈×C₆₇	#103	1	2	yes	—
91	C₈×C₆₇	C₂×SD₁₆	1	1	yes	characteristic
92	C₈×C₆₇	C₂×Q₁₆	1	1	yes	characteristic
93	C₆₇⋊C₈	—	8	2	no	—
94	C₂²×C₆₇	—	2	1	no	—
95	C₂²×C₆₇	—	2	1	no	—
96	C₂²×C₆₇	#332	1	2	yes	—
97	C₂²×C₆₇	C₂×#102	1	1	yes	characteristic
98	C₄×C₆₇	—	2	1	no	—
99	C₄×C₆₇	—	2	1	no	—
100	C₄×C₆₇	—	2	1	no	—
101	C₄×C₆₇	#332	1	2	yes	—
102	C₄×C₆₇	C₂×#101	1	1	yes	characteristic
103	C₄×C₆₇	C₂×#103	1	1	yes	characteristic
104	C₄×C₆₇	#508	1	2	yes	—
105	Dic₆₇	—	8	2	no	—
106	Dic₆₇	—	8	2	no	—
107	#11939	—	67	1	no	maximal, Sylow
108	C₂×C₆₇	—	2	2	no	—
109	C₂×C₆₇	C₂×#332	1	1	yes	characteristic
110	C₂×C₆₇	#2138	1	1	yes	characteristic
111	C₂×C₆₇	#2148	1	1	yes	characteristic
112	#3160	—	67	1	no	—
113	C₂×C₄⋊C₁₆	D₆₇	1	1	yes	characteristic
114	#1564	—	67	1	no	—
115	#1365	—	67	2	no	—
116	#1364	—	67	2	no	—
117	C₆₇	#11939	1	1	yes	characteristic, complemented, Sylow
118	#533	—	134	1	no	—
119	C₂²×C₁₆	—	2	1	no	—
120	#500	—	67	1	no	—
121	#500	—	67	1	no	—
122	#499	—	67	2	no	—
123	#499	—	134	1	no	—
124	#498	—	134	1	no	—
125	#431	—	134	1	no	—
126	#427	—	67	2	no	—
127	#427	—	134	1	no	—
128	C₂×C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
129	C₄⋊C₁₆	—	2	1	no	—
130	C₄⋊C₁₆	C₂×D₆₇	1	2	yes	—
131	#335	—	67	2	no	—
132	#334	—	134	1	no	—
133	C₂²×C₈	C₄×D₆₇	1	1	yes	characteristic
134	C₂²×C₈	C₂²⋊D₆₇	1	1	yes	characteristic
135	C₄⋊Q₈	—	134	1	no	—
136	C₄⋊Q₈	—	134	1	no	—
137	#125	—	134	2	no	—
138	C₂²⋊Q₈	—	268	1	no	—
139	C₂²⋊Q₈	—	268	1	no	—
140	C₄×Q₈	—	268	1	no	—
141	C₄×Q₈	—	268	1	no	—
142	#117	—	134	2	no	—
143	C₂×C₄²	D₂₆₈	1	1	yes	characteristic
144	C₂×C₁₆	—	2	2	no	—
145	C₂×C₁₆	—	4	2	no	—
146	#107	—	67	2	no	—
147	#107	—	67	2	no	—
148	#107	—	134	1	no	—
149	#107	—	134	1	no	—
150	#106	—	134	2	no	—
151	#106	—	134	2	no	—
152	C₄⋊C₈	—	134	2	no	—
153	C₂²⋊C₈	—	268	1	no	—
154	C₄×C₈	—	268	1	no	—
155	C₄×C₈	C₄×D₆₇	1	1	yes	characteristic
156	C₄×C₈	D₂₆₈	1	1	yes	characteristic
157	C₄×C₈	C₂²⋊D₆₇	1	1	yes	characteristic
158	C₄×C₈	C₂²×D₆₇	1	1	yes	characteristic
159	C₂×Q₈	—	268	1	no	—
160	C₂×Q₈	—	268	1	no	—
161	C₂²×C₄	(C₂×D₆₇)⋊C₄	1	1	yes	characteristic
162	C₂²×C₄	D₅₃₆	1	1	yes	characteristic
163	C₂²×C₄	#11761617	1	1	yes	characteristic
164	C₂×C₈	—	2	2	no	—
165	C₂×C₈	—	268	2	no	—
166	C₂×C₈	(C₂×D₆₇)⋊C₄	1	1	yes	characteristic
167	C₂×C₈	(C₂×D₆₇)⋊C₄	1	1	yes	characteristic
168	C₂×C₈	#11761616	1	2	yes	—
169	C₂×C₈	#11761618	1	2	yes	—
170	C₂×C₈	C₂×C₄×D₆₇	1	1	yes	characteristic
171	C₂×C₈	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
172	C₄⋊C₄	—	268	2	no	—
173	C₄⋊C₄	—	268	2	no	—
174	C₄⋊C₄	—	268	1	no	—
175	C₄⋊C₄	—	268	1	no	—
176	C₄⋊C₄	—	134	2	no	—
177	C₄⋊C₄	—	134	2	no	—
178	C₂²⋊C₄	—	268	2	no	—
179	C₄²	—	268	2	no	—
180	C₄²	(C₂×D₆₇)⋊C₄	1	1	yes	characteristic
181	C₄²	D₅₃₆	1	1	yes	characteristic
182	C₄²	#11761617	1	1	yes	characteristic
183	C₄²	C₂×D₂₆₈	1	1	yes	characteristic
184	C₁₆	—	4	2	no	—
185	C₂³	#14523965	1	1	yes	characteristic
186	Q₈	—	536	2	no	—
187	Q₈	—	536	2	no	—
188	C₂×C₄	—	2	2	no	—
189	C₂×C₄	—	4	1	no	—
190	C₂×C₄	—	268	2	no	—
191	C₂×C₄	—	268	2	no	—
192	C₂×C₄	#14523961	1	2	yes	—
193	C₂×C₄	#14523965	1	1	yes	characteristic
194	C₂×C₄	#14523965	1	1	yes	characteristic
195	C₂×C₄	#14523965	1	1	yes	characteristic
196	C₂×C₄	#14523966	1	2	yes	—
197	C₂×C₄	C₂×(C₂×D₆₇)⋊C₄	1	1	yes	characteristic
198	C₂×C₄	C₂×D₅₃₆	1	1	yes	characteristic
199	C₂×C₄	C₂×#11761617	1	1	yes	characteristic
200	C₈	—	2	1	no	—
201	C₈	—	2	1	no	—
202	C₈	—	536	2	no	—
203	C₈	#14523966	1	2	yes	—
204	C₈	C₂×#11761616	1	1	yes	characteristic
205	C₈	C₂×#11761618	1	1	yes	characteristic
206	C₂²	—	2	1	no	—
207	C₂²	—	2	1	no	—
208	C₂²	#16863141	1	2	yes	—
209	C₂²	C₂×#14523965	1	1	yes	characteristic
210	C₄	—	2	1	no	—
211	C₄	—	2	1	no	—
212	C₄	—	2	1	no	—
213	C₄	—	536	2	no	—
214	C₄	—	536	2	no	—
215	C₄	#16863141	1	2	yes	—
216	C₄	C₂×#14523961	1	1	yes	characteristic
217	C₄	C₂×#14523966	1	1	yes	characteristic
218	C₄	#16863828	1	2	yes	—
219	C₂	—	2	2	no	—
220	C₂	C₂×#16863141	1	1	yes	characteristic
221	C₂	#22561917	1	1	yes	characteristic
222	C₂	#22561959	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	#14523965
commutator	C₂×C₈×C₆₇	C₂²×C₄
Frattini	C₄×C₈	C₂²×D₆₇
Fitting	C₂×C₄⋊C₁₆×C₆₇	C₂
socle	C₂²×C₆₇	C₂×#102
prime radical	C₂³×C₆₇	#102
cosocle	C₄×C₈×C₆₇	C₂³
hypercenter	C₂×C₄⋊C₁₆	D₆₇
hypocenter	C₆₇	#11939
LCS	G ▷ C₂×C₈×C₆₇ ▷ C₂×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×C₄⋊C₁₆

Complements

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

N	H ≅ G/N	type
3. #22558266	C₂	semidirect
4. #22558263	C₂	semidirect
117. C₆₇	#11939	semidirect

Maximal subgroups

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

subgroup	index
#22570723	2
#22558957	2
#22558266	2
#22558263	2
C₂×C₄⋊C₁₆×C₆₇	2
#11939	67