Subgroups — #26102844

Order 12032.

344 conjugacy classes of subgroups.

The subgroup lattice has 212 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20243505	C₂	1	1	yes	maximal, characteristic
2	C₄×#13093935	C₂	1	1	yes	maximal, characteristic
3	C₄₇×#1744	C₂	1	1	yes	maximal, characteristic
4	C₂×C₄⋊(C₄₇⋊C₈)	C₄	1	2	yes	—
5	C₂×#13093935	—	2	1	no	—
6	C₂×#13093935	C₄	1	2	yes	—
7	C₂×C₄×C₄₇⋊C₈	—	2	1	no	—
8	#15688878	—	2	1	no	—
9	#15688878	—	2	1	no	—
10	C₂×C₄×C₈×C₄₇	—	2	1	no	—
11	C₄³×C₄₇	C₂²	1	1	yes	characteristic
12	C₄₇×#336	—	2	1	no	—
13	C₄₇×#336	C₄	1	2	yes	—
14	C₂²×C₄₇⋊C₈	—	2	2	no	—
15	C₂²×C₄₇⋊C₈	—	2	2	no	—
16	C₄⋊(C₄₇⋊C₈)	—	4	2	no	—
17	#13093935	—	2	4	no	—
18	#13093935	—	2	2	no	—
19	#13093935	C₈	1	4	yes	complemented
20	C₄×C₄₇⋊C₈	—	2	4	no	—
21	C₂²×C₈×C₄₇	—	2	2	no	—
22	C₂×C₄²×C₄₇	C₂×C₄	1	1	yes	characteristic
23	C₂×C₄²×C₄₇	C₂×C₄	1	1	yes	characteristic
24	C₂×C₄²×C₄₇	C₂×C₄	1	1	yes	characteristic
25	C₂×C₄²×C₄₇	D₄	1	1	yes	characteristic
26	C₂×C₄²×C₄₇	D₄	1	1	yes	characteristic
27	C₂×C₄²×C₄₇	D₄	1	1	yes	characteristic
28	C₂×C₄²×C₄₇	Q₈	1	1	yes	characteristic
29	C₄×C₈×C₄₇	—	2	4	no	—
30	C₂²×C₄×C₄₇	C₄²	1	1	yes	characteristic
31	C₂²×C₄×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
32	C₂²×C₄×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
33	C₂²×C₄×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
34	C₂²×C₄×C₄₇	C₄⋊C₄	1	1	yes	characteristic
35	C₂²×C₄×C₄₇	C₄⋊C₄	1	1	yes	characteristic
36	C₂²×C₄×C₄₇	C₄⋊C₄	1	1	yes	characteristic
37	C₂×C₈×C₄₇	—	2	4	no	—
38	C₂×C₈×C₄₇	—	4	2	no	—
39	C₂×C₈×C₄₇	—	4	2	no	—
40	C₄²×C₄₇	—	2	1	no	—
41	C₄²×C₄₇	—	2	1	no	—
42	C₄²×C₄₇	—	2	2	no	—
43	C₄²×C₄₇	—	2	2	no	—
44	C₄²×C₄₇	—	2	2	no	—
45	C₄²×C₄₇	—	2	2	no	—
46	C₄²×C₄₇	—	4	1	no	—
47	C₄²×C₄₇	C₂×C₈	1	2	yes	—
48	C₄²×C₄₇	M₁₆	1	2	yes	—
49	C₂×C₄₇⋊C₈	—	2	4	no	—
50	C₂×C₄₇⋊C₈	—	2	4	no	—
51	C₂×C₄₇⋊C₈	—	2	4	no	—
52	C₂×C₄₇⋊C₈	—	4	2	no	—
53	C₂×C₄₇⋊C₈	—	4	2	no	—
54	C₂×C₄₇⋊C₈	—	4	2	no	—
55	C₂³×C₄₇	#95	1	1	yes	characteristic
56	C₂×C₄×C₄₇	—	2	1	no	—
57	C₂×C₄×C₄₇	—	2	1	no	—
58	C₂×C₄×C₄₇	—	2	1	no	—
59	C₂×C₄×C₄₇	—	2	1	no	—
60	C₂×C₄×C₄₇	—	2	1	no	—
61	C₂×C₄×C₄₇	—	2	1	no	—
62	C₂×C₄×C₄₇	—	2	1	no	—
63	C₂×C₄×C₄₇	—	2	1	no	—
64	C₂×C₄×C₄₇	—	2	1	no	—
65	C₂×C₄×C₄₇	—	2	1	no	—
66	C₂×C₄×C₄₇	—	2	1	no	—
67	C₂×C₄×C₄₇	—	2	1	no	—
68	C₂×C₄×C₄₇	—	2	1	no	—
69	C₂×C₄×C₄₇	C₄×C₈	1	1	yes	characteristic
70	C₂×C₄×C₄₇	#97	1	1	yes	characteristic
71	C₂×C₄×C₄₇	C₂²⋊C₈	1	2	yes	—
72	C₂×C₄×C₄₇	#100	1	1	yes	characteristic
73	C₂×C₄×C₄₇	#101	1	1	yes	characteristic
74	C₂×C₄×C₄₇	#104	1	2	yes	—
75	C₂×C₄×C₄₇	#104	1	2	yes	—
76	C₂×C₄×C₄₇	#104	1	2	yes	—
77	C₂×C₄×C₄₇	C₄⋊C₈	1	2	yes	—
78	C₂×C₄×C₄₇	#108	1	2	yes	—
79	C₈×C₄₇	—	4	4	no	—
80	C₄₇⋊C₈	—	2	8	no	—
81	C₄₇⋊C₈	—	8	2	no	—
82	#7030	—	47	1	no	maximal, Sylow
83	C₂²×C₄₇	#336	1	1	yes	characteristic
84	C₂²×C₄₇	#339	1	1	yes	characteristic
85	C₂²×C₄₇	#339	1	1	yes	characteristic
86	C₂²×C₄₇	#339	1	1	yes	characteristic
87	C₂²×C₄₇	#341	1	1	yes	characteristic
88	C₂²×C₄₇	#343	1	1	yes	characteristic
89	C₂²×C₄₇	#344	1	1	yes	characteristic
90	C₄×C₄₇	—	2	1	no	—
91	C₄×C₄₇	—	2	1	no	—
92	C₄×C₄₇	—	2	2	no	—
93	C₄×C₄₇	—	2	2	no	—
94	C₄×C₄₇	—	2	2	no	—
95	C₄×C₄₇	—	2	2	no	—
96	C₄×C₄₇	—	4	1	no	—
97	C₄×C₄₇	#329	1	2	yes	—
98	C₄×C₄₇	#330	1	2	yes	—
99	#1831	—	47	1	no	—
100	#1744	D₄₇	1	1	yes	characteristic
101	C₄×#97	—	47	1	no	—
102	C₂×C₄₇	#1276	1	1	yes	characteristic
103	C₂×C₄₇	#1278	1	1	yes	characteristic
104	C₂×C₄₇	#1283	1	1	yes	characteristic
105	C₂×C₄₇	#1295	1	1	yes	characteristic
106	C₂×C₄₇	#1299	1	1	yes	characteristic
107	C₂×C₄₇	#1300	1	1	yes	characteristic
108	C₂×C₄₇	#1302	1	1	yes	characteristic
109	C₂×C₄⋊C₈	—	47	2	no	—
110	C₂×#97	—	47	2	no	—
111	C₂×#97	—	94	1	no	—
112	C₂×C₄×C₈	—	2	1	no	—
113	C₂×C₄×C₈	—	94	1	no	—
114	C₄³	C₂×D₄₇	1	1	yes	characteristic
115	#336	—	2	1	no	—
116	#336	—	94	1	no	—
117	#336	—	94	1	no	—
118	#336	Dic₄₇	1	2	yes	—
119	C₄₇	#7030	1	1	yes	characteristic, complemented, Sylow
120	C₂²×C₈	—	2	2	no	—
121	C₂²×C₈	—	94	2	no	—
122	C₂²×C₈	—	94	2	no	—
123	C₂×C₄²	Dic₉₄	1	1	yes	characteristic
124	C₂×C₄²	C₄×D₄₇	1	1	yes	characteristic
125	C₂×C₄²	C₄×D₄₇	1	1	yes	characteristic
126	C₂×C₄²	D₁₈₈	1	1	yes	characteristic
127	C₂×C₄²	C₂×Dic₄₇	1	1	yes	characteristic
128	C₂×C₄²	C₂²⋊D₄₇	1	1	yes	characteristic
129	C₂×C₄²	C₂²⋊D₄₇	1	1	yes	characteristic
130	C₄⋊C₈	—	188	2	no	—
131	#97	—	47	4	no	—
132	#97	—	94	4	no	—
133	#97	—	94	2	no	—
134	C₄×C₈	—	2	4	no	—
135	C₄×C₈	—	94	4	no	—
136	C₂²×C₄	C₄×Dic₄₇	1	1	yes	characteristic
137	C₂²×C₄	Dic₄₇⋊C₄	1	1	yes	characteristic
138	C₂²×C₄	Dic₄₇⋊C₄	1	1	yes	characteristic
139	C₂²×C₄	C₄⋊Dic₄₇	1	1	yes	characteristic
140	C₂²×C₄	(C₂×D₄₇)⋊C₄	1	1	yes	characteristic
141	C₂²×C₄	(C₂×D₄₇)⋊C₄	1	1	yes	characteristic
142	C₂²×C₄	C₂²⋊Dic₄₇	1	1	yes	characteristic
143	C₂×C₈	—	2	4	no	—
144	C₂×C₈	—	4	2	no	—
145	C₂×C₈	—	4	2	no	—
146	C₂×C₈	—	94	4	no	—
147	C₂×C₈	—	94	4	no	—
148	C₂×C₈	—	94	4	no	—
149	C₂×C₈	—	188	2	no	—
150	C₂×C₈	—	188	2	no	—
151	C₂×C₈	—	188	2	no	—
152	C₄²	—	2	1	no	—
153	C₄²	—	2	1	no	—
154	C₄²	—	2	2	no	—
155	C₄²	—	2	2	no	—
156	C₄²	—	2	2	no	—
157	C₄²	—	2	2	no	—
158	C₄²	—	4	1	no	—
159	C₄²	C₈×D₄₇	1	2	yes	—
160	C₄²	#10625324	1	2	yes	—
161	C₂³	C₄₇⋊#95	1	1	yes	characteristic
162	C₂×C₄	—	2	1	no	—
163	C₂×C₄	—	2	1	no	—
164	C₂×C₄	—	2	1	no	—
165	C₂×C₄	—	2	1	no	—
166	C₂×C₄	—	2	1	no	—
167	C₂×C₄	—	2	1	no	—
168	C₂×C₄	—	2	1	no	—
169	C₂×C₄	—	2	1	no	—
170	C₂×C₄	—	2	1	no	—
171	C₂×C₄	—	2	1	no	—
172	C₂×C₄	—	2	1	no	—
173	C₂×C₄	—	2	1	no	—
174	C₂×C₄	—	2	1	no	—
175	C₂×C₄	C₈×Dic₄₇	1	1	yes	characteristic
176	C₂×C₄	#13093934	1	1	yes	characteristic
177	C₂×C₄	(C₂×D₄₇)⋊C₈	1	2	yes	—
178	C₂×C₄	#13093940	1	1	yes	characteristic
179	C₂×C₄	#13093942	1	1	yes	characteristic
180	C₂×C₄	#13093950	1	2	yes	—
181	C₂×C₄	#13093951	1	2	yes	—
182	C₂×C₄	#13093952	1	2	yes	—
183	C₂×C₄	Dic₄₇⋊C₈	1	2	yes	—
184	C₂×C₄	#13093960	1	2	yes	—
185	C₈	—	4	4	no	—
186	C₈	—	94	8	no	—
187	C₈	—	376	2	no	—
188	C₂²	#15688877	1	1	yes	characteristic
189	C₂²	#15688882	1	1	yes	characteristic
190	C₂²	#15688883	1	1	yes	characteristic
191	C₂²	#15688884	1	1	yes	characteristic
192	C₂²	#15688888	1	1	yes	characteristic
193	C₂²	#15688891	1	1	yes	characteristic
194	C₂²	#15688893	1	1	yes	characteristic
195	C₄	—	2	2	no	—
196	C₄	—	2	2	no	—
197	C₄	—	2	1	no	—
198	C₄	—	2	1	no	—
199	C₄	—	2	2	no	—
200	C₄	—	2	2	no	—
201	C₄	—	4	1	no	—
202	C₄	#15688862	1	2	yes	—
203	C₄	#15688864	1	2	yes	—
204	C₂	#20241220	1	1	yes	characteristic
205	C₂	#20241225	1	1	yes	characteristic
206	C₂	#20241237	1	1	yes	characteristic
207	C₂	#20241270	1	1	yes	characteristic
208	C₂	#20241279	1	1	yes	characteristic
209	C₂	#20241281	1	1	yes	characteristic
210	C₂	#20241288	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	(C₂×D₄₇)⋊C₄
commutator	C₂×C₄×C₄₇	C₄×C₈
Frattini	C₄³	C₂×D₄₇
Fitting	C₄₇×#1744	C₂
socle, prime radical	C₂³×C₄₇	#95
cosocle	C₄³×C₄₇	C₂²
hypercenter	#1744	D₄₇
hypocenter	C₄₇	#7030
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₂²×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂²×C₄ ◁ C₄³ ◁ #1744

Complements

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

N	H ≅ G/N	type
19. #13093935	C₈	semidirect
119. C₄₇	#7030	semidirect

Maximal subgroups

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

subgroup	index
#20243505	2
C₄×#13093935	2
C₄₇×#1744	2
#7030	47