Subgroups — #30767129

Order 17152.

310 conjugacy classes of subgroups.

The subgroup lattice has 232 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	C₂×#16863149	C₂	1	1	yes	maximal, characteristic
3	#22558266	C₂	1	2	yes	maximal, complemented
4	#22558263	C₂	1	2	yes	maximal, complemented
5	C₆₇×#2150	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	#16863434	—	2	1	no	—
13	#16863434	C₄	1	2	yes	—
14	C₂×C₄⋊(C₆₇⋊C₈)	—	2	1	no	—
15	#16863149	—	2	1	no	—
16	#16863149	C₂²	1	2	yes	—
17	C₂×M₃₂×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	C₂²×C₆₇⋊C₈	—	4	1	no	—
22	#14524043	—	2	1	no	—
23	#14524043	—	2	1	no	—
24	#14524035	—	2	2	no	—
25	#14524022	—	4	1	no	—
26	#14524022	—	4	1	no	—
27	C₄×Dic₁₃₄	—	4	1	no	—
28	C₄×Dic₁₃₄	—	4	1	no	—
29	#14524002	—	2	2	no	—
30	#14523977	—	2	1	no	—
31	#14523977	—	2	1	no	—
32	#14523977	C₂×C₄	1	2	yes	—
33	#14523977	C₂×C₄	1	2	yes	—
34	#14523975	—	2	2	no	—
35	#14523975	—	2	2	no	—
36	C₄⋊(C₆₇⋊C₈)	—	2	2	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	M₃₂×C₆₇	—	4	2	no	—
42	C₂×C₁₆×C₆₇	—	2	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	C₂×C₆₇⋊C₈	—	8	2	no	—
51	C₄⋊Dic₆₇	—	2	2	no	—
52	C₄⋊Dic₆₇	—	2	2	no	—
53	C₄⋊Dic₆₇	—	4	1	no	—
54	C₄⋊Dic₆₇	—	4	1	no	—
55	Dic₆₇⋊C₄	—	4	2	no	—
56	Dic₆₇⋊C₄	—	4	2	no	—
57	C₂²⋊Dic₆₇	—	4	2	no	—
58	C₄×Dic₆₇	—	4	2	no	—
59	C₂²×C₄×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
60	C₂²×C₄×C₆₇	D₈	1	1	yes	characteristic
61	C₂²×C₄×C₆₇	SD₁₆	1	1	yes	characteristic
62	C₂×C₈×C₆₇	—	2	2	no	—
63	C₂×C₈×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
64	C₂×C₈×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
65	C₂×C₈×C₆₇	SD₁₆	1	2	yes	—
66	C₂×C₈×C₆₇	Q₁₆	1	2	yes	—
67	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
68	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
69	C₄²×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
70	C₄²×C₆₇	D₈	1	1	yes	characteristic
71	C₄²×C₆₇	SD₁₆	1	1	yes	characteristic
72	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
73	C₁₆×C₆₇	—	4	2	no	—
74	C₂³×C₆₇	#102	1	1	yes	characteristic
75	C₂×C₄×C₆₇	—	2	1	no	—
76	C₂×C₄×C₆₇	—	2	1	no	—
77	C₂×C₄×C₆₇	—	4	1	no	—
78	C₂×C₄×C₆₇	#101	1	2	yes	—
79	C₂×C₄×C₆₇	#102	1	1	yes	characteristic
80	C₂×C₄×C₆₇	#102	1	1	yes	characteristic
81	C₂×C₄×C₆₇	#102	1	1	yes	characteristic
82	C₂×C₄×C₆₇	#103	1	2	yes	—
83	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
84	C₂×C₄×C₆₇	C₂×D₈	1	1	yes	characteristic
85	C₂×C₄×C₆₇	C₂×SD₁₆	1	1	yes	characteristic
86	C₂×Dic₆₇	—	4	2	no	—
87	C₂×Dic₆₇	—	4	2	no	—
88	Dic₁₃₄	—	8	2	no	—
89	Dic₁₃₄	—	8	1	no	—
90	Dic₁₃₄	—	8	1	no	—
91	C₈×C₆₇	—	2	1	no	—
92	C₈×C₆₇	—	2	1	no	—
93	C₈×C₆₇	#103	1	2	yes	—
94	C₈×C₆₇	C₂×SD₁₆	1	1	yes	characteristic
95	C₈×C₆₇	C₂×Q₁₆	1	1	yes	characteristic
96	C₆₇⋊C₈	—	8	2	no	—
97	C₂²×C₆₇	—	2	1	no	—
98	C₂²×C₆₇	—	2	1	no	—
99	C₂²×C₆₇	#332	1	2	yes	—
100	C₂²×C₆₇	C₂×#102	1	1	yes	characteristic
101	C₄×C₆₇	—	2	1	no	—
102	C₄×C₆₇	—	2	1	no	—
103	C₄×C₆₇	—	2	1	no	—
104	C₄×C₆₇	#332	1	2	yes	—
105	C₄×C₆₇	C₂×#101	1	1	yes	characteristic
106	C₄×C₆₇	C₂×#103	1	1	yes	characteristic
107	C₄×C₆₇	#509	1	1	yes	characteristic
108	C₄×C₆₇	#510	1	1	yes	characteristic
109	Dic₆₇	—	8	2	no	—
110	Dic₆₇	—	8	2	no	—
111	#11942	—	67	1	no	maximal, Sylow
112	C₂×C₆₇	—	2	2	no	—
113	C₂×C₆₇	C₂×#332	1	1	yes	characteristic
114	C₂×C₆₇	#2139	1	1	yes	characteristic
115	C₂×C₆₇	#2148	1	1	yes	characteristic
116	#3160	—	67	1	no	—
117	#2150	D₆₇	1	1	yes	characteristic
118	C₂×#335	—	67	1	no	—
119	#1365	—	67	2	no	—
120	#1364	—	67	2	no	—
121	C₆₇	#11942	1	1	yes	characteristic, complemented, Sylow
122	#533	—	134	1	no	—
123	C₂×M₃₂	—	2	1	no	—
124	#500	—	67	1	no	—
125	#500	—	67	1	no	—
126	#499	—	67	2	no	—
127	#499	—	134	1	no	—
128	#498	—	134	1	no	—
129	#427	—	67	2	no	—
130	#427	—	134	1	no	—
131	C₂×C₄⋊C₈	—	134	1	no	—
132	C₂×C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
133	C₄⋊C₁₆	—	2	1	no	—
134	C₄⋊C₁₆	C₂×D₆₇	1	2	yes	—
135	#335	—	67	2	no	—
136	#335	—	134	1	no	—
137	C₂²×C₈	—	268	1	no	—
138	C₂²×C₈	C₄×D₆₇	1	1	yes	characteristic
139	C₂²×C₈	C₂²⋊D₆₇	1	1	yes	characteristic
140	C₄⋊Q₈	—	134	1	no	—
141	C₄⋊Q₈	—	134	1	no	—
142	#125	—	134	2	no	—
143	C₂²⋊Q₈	—	268	1	no	—
144	C₂²⋊Q₈	—	268	1	no	—
145	C₄×Q₈	—	268	1	no	—
146	C₄×Q₈	—	268	1	no	—
147	#117	—	134	2	no	—
148	C₂×C₄²	D₂₆₈	1	1	yes	characteristic
149	M₃₂	—	4	2	no	—
150	C₂×C₁₆	—	2	2	no	—
151	#107	—	67	2	no	—
152	#107	—	67	2	no	—
153	#107	—	134	1	no	—
154	#107	—	134	1	no	—
155	#106	—	134	2	no	—
156	#106	—	134	2	no	—
157	C₄⋊C₈	—	268	1	no	—
158	C₄⋊C₈	—	134	2	no	—
159	C₄×C₈	C₄×D₆₇	1	1	yes	characteristic
160	C₄×C₈	D₂₆₈	1	1	yes	characteristic
161	C₄×C₈	C₂²⋊D₆₇	1	1	yes	characteristic
162	C₄×C₈	C₂²×D₆₇	1	1	yes	characteristic
163	C₂×Q₈	—	268	1	no	—
164	C₂×Q₈	—	268	1	no	—
165	C₂²×C₄	(C₂×D₆₇)⋊C₄	1	1	yes	characteristic
166	C₂²×C₄	D₅₃₆	1	1	yes	characteristic
167	C₂²×C₄	#11761617	1	1	yes	characteristic
168	C₂×C₈	—	2	2	no	—
169	C₂×C₈	—	268	2	no	—
170	C₂×C₈	—	536	2	no	—
171	C₂×C₈	(C₂×D₆₇)⋊C₄	1	1	yes	characteristic
172	C₂×C₈	(C₂×D₆₇)⋊C₄	1	1	yes	characteristic
173	C₂×C₈	#11761616	1	2	yes	—
174	C₂×C₈	#11761618	1	2	yes	—
175	C₂×C₈	C₂×C₄×D₆₇	1	1	yes	characteristic
176	C₂×C₈	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
177	C₄⋊C₄	—	268	2	no	—
178	C₄⋊C₄	—	268	2	no	—
179	C₄⋊C₄	—	268	1	no	—
180	C₄⋊C₄	—	268	1	no	—
181	C₄⋊C₄	—	134	2	no	—
182	C₄⋊C₄	—	134	2	no	—
183	C₂²⋊C₄	—	268	2	no	—
184	C₄²	—	268	2	no	—
185	C₄²	(C₂×D₆₇)⋊C₄	1	1	yes	characteristic
186	C₄²	D₅₃₆	1	1	yes	characteristic
187	C₄²	#11761617	1	1	yes	characteristic
188	C₄²	C₂×D₂₆₈	1	1	yes	characteristic
189	C₁₆	—	4	2	no	—
190	C₂³	#14523965	1	1	yes	characteristic
191	Q₈	—	536	2	no	—
192	Q₈	—	536	1	no	—
193	Q₈	—	536	1	no	—
194	C₂×C₄	—	2	1	no	—
195	C₂×C₄	—	2	1	no	—
196	C₂×C₄	—	4	1	no	—
197	C₂×C₄	—	268	2	no	—
198	C₂×C₄	—	268	2	no	—
199	C₂×C₄	#14523961	1	2	yes	—
200	C₂×C₄	#14523965	1	1	yes	characteristic
201	C₂×C₄	#14523965	1	1	yes	characteristic
202	C₂×C₄	#14523965	1	1	yes	characteristic
203	C₂×C₄	#14523966	1	2	yes	—
204	C₂×C₄	C₂×(C₂×D₆₇)⋊C₄	1	1	yes	characteristic
205	C₂×C₄	C₂×D₅₃₆	1	1	yes	characteristic
206	C₂×C₄	C₂×#11761617	1	1	yes	characteristic
207	C₈	—	2	1	no	—
208	C₈	—	2	1	no	—
209	C₈	—	536	2	no	—
210	C₈	#14523966	1	2	yes	—
211	C₈	C₂×#11761616	1	1	yes	characteristic
212	C₈	C₂×#11761618	1	1	yes	characteristic
213	C₂²	—	2	1	no	—
214	C₂²	—	2	1	no	—
215	C₂²	#16863141	1	2	yes	—
216	C₂²	C₂×#14523965	1	1	yes	characteristic
217	C₄	—	2	1	no	—
218	C₄	—	2	1	no	—
219	C₄	—	2	1	no	—
220	C₄	—	536	2	no	—
221	C₄	—	536	2	no	—
222	C₄	#16863141	1	2	yes	—
223	C₄	C₂×#14523961	1	1	yes	characteristic
224	C₄	C₂×#14523966	1	1	yes	characteristic
225	C₄	#16863834	1	1	yes	characteristic
226	C₄	#16863839	1	1	yes	characteristic
227	C₂	—	2	2	no	—
228	C₂	C₂×#16863141	1	1	yes	characteristic
229	C₂	#22561922	1	1	yes	characteristic
230	C₂	#22561959	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×#14523965
commutator	C₂×C₈×C₆₇	C₂²×C₄
Frattini	C₄×C₈	C₂²×D₆₇
Fitting	C₆₇×#2150	C₂
socle	C₂²×C₆₇	C₂×#102
prime radical	C₂³×C₆₇	#102
cosocle	C₄×C₈×C₆₇	C₂³
hypercenter	#2150	D₆₇
hypocenter	C₆₇	#11942
LCS	G ▷ C₂×C₈×C₆₇ ▷ C₂×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ #2150

Complements

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

N	H ≅ G/N	type
3. #22558266	C₂	semidirect
4. #22558263	C₂	semidirect
121. C₆₇	#11942	semidirect

Maximal subgroups

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

subgroup	index
#22570723	2
C₂×#16863149	2
#22558266	2
#22558263	2
C₆₇×#2150	2
#11942	67