Subgroups — #30774433

Order 17152.

414 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	C₂×C₄×C₆₇⋊C₁₆	C₂	1	1	yes	maximal, characteristic
2	#22560369	C₂	1	1	yes	maximal, characteristic
3	#22558300	C₂	1	4	yes	maximal
4	C₂×#500×C₆₇	C₂	1	1	yes	maximal, characteristic
5	C₂²×C₆₇⋊C₁₆	C₂²	1	2	yes	—
6	C₂×#14523977	—	2	1	no	—
7	C₂×#14523977	C₂²	1	2	yes	—
8	#16863266	—	2	1	no	—
9	C₄×C₆₇⋊C₁₆	—	2	2	no	—
10	C₂×C₄⋊Q₈×C₆₇	—	2	1	no	—
11	#500×C₆₇	—	2	2	no	—
12	#500×C₆₇	C₄	1	4	yes	complemented
13	C₂×#107×C₆₇	—	2	1	no	—
14	C₂×#107×C₆₇	C₂²	1	2	yes	—
15	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
16	C₂×C₄⋊Dic₆₇	—	2	2	no	—
17	C₂×C₄⋊Dic₆₇	—	4	1	no	—
18	C₂×C₄⋊Dic₆₇	—	4	1	no	—
19	C₂×C₆₇⋊C₁₆	—	2	2	no	—
20	C₂×C₆₇⋊C₁₆	—	2	2	no	—
21	C₂×C₆₇⋊C₁₆	Q₈	1	4	yes	complemented
22	#14523977	—	2	4	no	—
23	#14523977	—	4	2	no	—
24	C₂²×Q₈×C₆₇	—	4	1	no	—
25	C₂²×C₈×C₆₇	D₄	1	1	yes	characteristic
26	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
27	C₄⋊Q₈×C₆₇	—	2	4	no	—
28	C₄⋊Q₈×C₆₇	—	4	2	no	—
29	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
30	C₂×C₄⋊C₄×C₆₇	—	4	1	no	—
31	C₂×C₄²×C₆₇	D₄	1	1	yes	characteristic
32	#107×C₆₇	—	2	4	no	—
33	#107×C₆₇	—	2	2	no	—
34	#107×C₆₇	C₂×C₄	1	4	yes	—
35	C₄×C₈×C₆₇	C₂×C₄	1	2	yes	—
36	C₄×C₈×C₆₇	D₄	1	2	yes	—
37	C₂²×Dic₆₇	—	4	2	no	—
38	C₄⋊Dic₆₇	—	4	2	no	—
39	C₄⋊Dic₆₇	—	4	2	no	—
40	C₄⋊Dic₆₇	—	8	2	no	—
41	C₄⋊Dic₆₇	—	8	2	no	—
42	C₆₇⋊C₁₆	—	2	4	no	—
43	C₂×Q₈×C₆₇	—	4	4	no	—
44	C₂×Q₈×C₆₇	—	8	2	no	—
45	C₂×Q₈×C₆₇	—	8	2	no	—
46	C₂²×C₄×C₆₇	—	4	2	no	—
47	C₂²×C₄×C₆₇	D₈	1	1	yes	characteristic
48	C₂²×C₄×C₆₇	D₈	1	1	yes	characteristic
49	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
50	C₂×C₈×C₆₇	C₂²⋊C₄	1	2	yes	—
51	C₂×C₈×C₆₇	SD₁₆	1	2	yes	—
52	C₂×C₈×C₆₇	Q₁₆	1	2	yes	—
53	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
54	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
55	C₂×C₈×C₆₇	C₂×Q₈	1	2	yes	—
56	C₂×C₈×C₆₇	#41	1	2	yes	—
57	C₄⋊C₄×C₆₇	—	2	4	no	—
58	C₄⋊C₄×C₆₇	—	4	4	no	—
59	C₄⋊C₄×C₆₇	—	4	2	no	—
60	C₄²×C₆₇	C₂²⋊C₄	1	2	yes	—
61	C₄²×C₆₇	SD₁₆	1	2	yes	—
62	C₂³×C₆₇	C₂×D₈	1	1	yes	characteristic
63	Q₈×C₆₇	—	8	8	no	—
64	C₂×C₄×C₆₇	—	4	4	no	—
65	C₂×C₄×C₆₇	—	8	2	no	—
66	C₂×C₄×C₆₇	—	8	2	no	—
67	C₂×C₄×C₆₇	#102	1	2	yes	—
68	C₂×C₄×C₆₇	#102	1	2	yes	—
69	C₂×C₄×C₆₇	D₁₆	1	2	yes	—
70	C₂×C₄×C₆₇	SD₃₂	1	2	yes	—
71	C₂×C₄×C₆₇	SD₃₂	1	2	yes	—
72	C₂×C₄×C₆₇	Q₃₂	1	2	yes	—
73	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
74	C₂×C₄×C₆₇	#124	1	1	yes	characteristic
75	C₂×C₄×C₆₇	C₄⋊Q₈	1	1	yes	characteristic
76	C₂×C₄×C₆₇	C₂×SD₁₆	1	1	yes	characteristic
77	C₂×C₄×C₆₇	C₂×SD₁₆	1	1	yes	characteristic
78	C₂×C₄×C₆₇	C₂×Q₁₆	1	1	yes	characteristic
79	C₂×Dic₆₇	—	8	2	no	—
80	C₂×Dic₆₇	—	8	2	no	—
81	C₂×Dic₆₇	—	8	2	no	—
82	C₈×C₆₇	#103	1	4	yes	—
83	C₈×C₆₇	C₄×Q₈	1	2	yes	—
84	C₈×C₆₇	C₂²⋊Q₈	1	2	yes	—
85	C₂²×C₆₇	C₂×#102	1	1	yes	characteristic
86	C₂²×C₆₇	#486	1	1	yes	characteristic
87	C₂²×C₆₇	#500	1	1	yes	characteristic
88	C₂²×C₆₇	C₂×D₁₆	1	1	yes	characteristic
89	C₂²×C₆₇	C₂×SD₃₂	1	1	yes	characteristic
90	C₂²×C₆₇	C₂×SD₃₂	1	1	yes	characteristic
91	C₂²×C₆₇	C₂×Q₃₂	1	1	yes	characteristic
92	C₄×C₆₇	—	8	4	no	—
93	C₄×C₆₇	#357	1	4	yes	—
94	C₄×C₆₇	#358	1	4	yes	—
95	C₄×C₆₇	#391	1	1	yes	characteristic
96	C₄×C₆₇	C₂×#103	1	1	yes	characteristic
97	C₄×C₆₇	#487	1	1	yes	characteristic
98	C₄×C₆₇	#498	1	1	yes	characteristic
99	Dic₆₇	—	16	2	no	—
100	#13282	—	67	1	no	maximal, Sylow
101	C₂×C₆₇	#1990	1	1	yes	characteristic
102	C₂×C₆₇	C₂×#357	1	1	yes	characteristic
103	C₂×C₆₇	C₂×#358	1	1	yes	characteristic
104	C₂×C₆₇	#2240	1	1	yes	characteristic
105	C₂×C₆₇	#2241	1	1	yes	characteristic
106	C₂×C₆₇	#2252	1	1	yes	characteristic
107	C₂×C₆₇	#2254	1	1	yes	characteristic
108	C₂×#500	D₆₇	1	1	yes	characteristic
109	C₂×C₄×C₁₆	—	67	1	no	—
110	#1845	—	67	1	no	—
111	#1380	—	67	4	no	—
112	C₆₇	#13282	1	1	yes	characteristic, complemented, Sylow
113	C₂×C₄⋊Q₈	—	2	1	no	—
114	C₂²×C₁₆	—	67	2	no	—
115	#500	—	2	2	no	—
116	#500	Dic₆₇	1	4	yes	complemented
117	C₂×#107	—	2	1	no	—
118	C₂×#107	—	67	2	no	—
119	C₂×#107	—	134	1	no	—
120	C₂×#107	C₂×D₆₇	1	2	yes	—
121	C₂×C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
122	#384	—	134	1	no	—
123	C₄×C₁₆	—	134	2	no	—
124	C₂²×Q₈	—	4	1	no	—
125	C₂²×C₈	C₂²⋊D₆₇	1	1	yes	characteristic
126	C₂²×C₈	C₂²×D₆₇	1	1	yes	characteristic
127	C₄⋊Q₈	—	2	4	no	—
128	C₄⋊Q₈	—	4	2	no	—
129	C₂×C₄⋊C₄	—	2	2	no	—
130	C₂×C₄⋊C₄	—	4	1	no	—
131	C₂×C₄⋊C₄	—	268	1	no	—
132	C₂×C₄⋊C₄	—	268	1	no	—
133	C₂×C₄⋊C₄	—	134	2	no	—
134	C₂×C₄²	C₂²⋊D₆₇	1	1	yes	characteristic
135	C₂×C₁₆	—	67	4	no	—
136	C₂×C₁₆	—	134	2	no	—
137	C₂×C₁₆	—	134	2	no	—
138	#107	—	2	4	no	—
139	#107	—	2	2	no	—
140	#107	—	268	2	no	—
141	#107	—	134	4	no	—
142	#107	C₂×Dic₆₇	1	4	yes	—
143	C₄×C₈	C₂×Dic₆₇	1	2	yes	—
144	C₄×C₈	C₂²⋊D₆₇	1	2	yes	—
145	C₂×Q₈	—	4	4	no	—
146	C₂×Q₈	—	8	2	no	—
147	C₂×Q₈	—	8	2	no	—
148	C₂²×C₄	—	4	2	no	—
149	C₂²×C₄	—	268	2	no	—
150	C₂²×C₄	#11761613	1	1	yes	characteristic
151	C₂²×C₄	#11761613	1	1	yes	characteristic
152	C₂²×C₄	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
153	C₂×C₈	C₂²⋊Dic₆₇	1	2	yes	—
154	C₂×C₈	#11761615	1	2	yes	—
155	C₂×C₈	#11761618	1	2	yes	—
156	C₂×C₈	C₂²×Dic₆₇	1	1	yes	characteristic
157	C₂×C₈	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
158	C₂×C₈	Q₈×D₆₇	1	2	yes	—
159	C₂×C₈	#11761629	1	2	yes	—
160	C₄⋊C₄	—	2	4	no	—
161	C₄⋊C₄	—	4	4	no	—
162	C₄⋊C₄	—	4	2	no	—
163	C₄⋊C₄	—	268	2	no	—
164	C₄⋊C₄	—	268	2	no	—
165	C₄⋊C₄	—	536	2	no	—
166	C₄⋊C₄	—	536	2	no	—
167	C₄²	C₂²⋊Dic₆₇	1	2	yes	—
168	C₄²	#11761616	1	2	yes	—
169	C₁₆	—	134	4	no	—
170	C₂³	C₂×#11761613	1	1	yes	characteristic
171	Q₈	—	8	8	no	—
172	C₂×C₄	—	4	4	no	—
173	C₂×C₄	—	8	2	no	—
174	C₂×C₄	—	8	2	no	—
175	C₂×C₄	—	536	2	no	—
176	C₂×C₄	—	536	2	no	—
177	C₂×C₄	—	536	2	no	—
178	C₂×C₄	#14523964	1	2	yes	—
179	C₂×C₄	#14523964	1	2	yes	—
180	C₂×C₄	#14523984	1	2	yes	—
181	C₂×C₄	#14523986	1	2	yes	—
182	C₂×C₄	#14523987	1	2	yes	—
183	C₂×C₄	#14523989	1	2	yes	—
184	C₂×C₄	C₂×C₂²⋊Dic₆₇	1	1	yes	characteristic
185	C₂×C₄	#14524032	1	1	yes	characteristic
186	C₂×C₄	#14524041	1	1	yes	characteristic
187	C₂×C₄	C₂×#11761615	1	1	yes	characteristic
188	C₂×C₄	C₂×#11761616	1	1	yes	characteristic
189	C₂×C₄	C₂×#11761618	1	1	yes	characteristic
190	C₈	#14523967	1	4	yes	—
191	C₈	Q₈×Dic₆₇	1	2	yes	—
192	C₈	#14524019	1	2	yes	—
193	C₂²	C₂×#14523964	1	1	yes	characteristic
194	C₂²	#16863744	1	1	yes	characteristic
195	C₂²	#16863799	1	1	yes	characteristic
196	C₂²	C₂×#14523984	1	1	yes	characteristic
197	C₂²	C₂×#14523986	1	1	yes	characteristic
198	C₂²	C₂×#14523987	1	1	yes	characteristic
199	C₂²	C₂×#14523989	1	1	yes	characteristic
200	C₄	—	8	4	no	—
201	C₄	—	1072	2	no	—
202	C₄	#16863197	1	4	yes	—
203	C₄	#16863200	1	4	yes	—
204	C₄	#16863303	1	1	yes	characteristic
205	C₄	C₂×#14523967	1	1	yes	characteristic
206	C₄	#16863748	1	1	yes	characteristic
207	C₄	#16863792	1	1	yes	characteristic
208	C₂	#22561171	1	1	yes	characteristic
209	C₂	C₂×#16863197	1	1	yes	characteristic
210	C₂	C₂×#16863200	1	1	yes	characteristic
211	C₂	#22562434	1	1	yes	characteristic
212	C₂	#22562438	1	1	yes	characteristic
213	C₂	#22562482	1	1	yes	characteristic
214	C₂	#22562489	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂×#11761613
commutator	C₂×C₈×C₆₇	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₆₇
Fitting	C₂×#500×C₆₇	C₂
socle, prime radical	C₂³×C₆₇	C₂×D₈
cosocle	C₂²×C₈×C₆₇	C₂³
hypercenter	C₂×#500	D₆₇
hypocenter	C₆₇	#13282
LCS	G ▷ C₂×C₈×C₆₇ ▷ C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×#500

Complements

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

N	H ≅ G/N	type
12. #500×C₆₇	C₄	semidirect
21. C₂×C₆₇⋊C₁₆	Q₈	semidirect
112. C₆₇	#13282	semidirect
116. #500	Dic₆₇	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄×C₆₇⋊C₁₆	2
#22560369	2
#22558300	2
C₂×#500×C₆₇	2
#13282	67