Subgroups — #26138054

Order 12032.

1422 conjugacy classes of subgroups.

The subgroup lattice has 180 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×C₄⋊Dic₄₇	C₂	1	2	yes	maximal
2	#20241645	C₂	1	8	yes	maximal
3	C₄×C₄₇⋊#95	C₂	1	4	yes	maximal
4	C₂×C₄₇×#376	C₂	1	1	yes	maximal, characteristic
5	C₂²×C₄⋊Dic₄₇	C₂²	1	4	yes	—
6	C₂²×C₄×Dic₄₇	C₂²	1	4	yes	—
7	C₄×C₄⋊Dic₄₇	—	2	8	no	—
8	C₂×C₄₇⋊#95	C₂²	1	16	yes	—
9	C₂×C₄₇⋊#95	C₂²	1	4	yes	—
10	C₂²×C₄²×C₄₇	C₂²	1	2	yes	—
11	C₂²×C₄²×C₄₇	C₂²	1	1	yes	characteristic
12	C₄₇×#376	C₄	1	8	yes	complemented
13	C₂×#95×C₄₇	C₂²	1	4	yes	—
14	C₂³×Dic₄₇	C₂³	1	8	yes	—
15	C₂×C₄⋊Dic₄₇	—	2	8	no	—
16	C₂×C₄⋊Dic₄₇	—	2	16	no	—
17	C₂×C₄×Dic₄₇	—	2	8	no	—
18	C₂×C₄×Dic₄₇	—	2	16	no	—
19	C₄₇⋊#95	—	2	8	no	—
20	C₄₇⋊#95	—	2	32	no	—
21	C₂³×C₄×C₄₇	C₂³	1	4	yes	—
22	C₂³×C₄×C₄₇	C₂³	1	2	yes	—
23	C₂³×C₄×C₄₇	C₂³	1	1	yes	characteristic
24	C₂×C₄²×C₄₇	—	2	8	no	—
25	C₂×C₄²×C₄₇	C₂×C₄	1	8	yes	—
26	C₂×C₄²×C₄₇	C₂×C₄	1	4	yes	—
27	C₂×C₄²×C₄₇	D₄	1	4	yes	—
28	C₂×C₄²×C₄₇	Q₈	1	4	yes	—
29	#95×C₄₇	C₂×C₄	1	16	yes	—
30	C₂⁴×C₄₇	C₂⁴	1	1	yes	characteristic
31	C₂²×C₄×C₄₇	—	2	8	no	—
32	C₂²×C₄×C₄₇	—	2	8	no	—
33	C₂²×C₄×C₄₇	—	2	16	no	—
34	C₂²×C₄×C₄₇	C₂²×C₄	1	8	yes	—
35	C₂²×C₄×C₄₇	C₂²×C₄	1	4	yes	—
36	C₂²×C₄×C₄₇	C₂²×C₄	1	2	yes	—
37	C₂²×C₄×C₄₇	C₂×D₄	1	4	yes	—
38	C₂²×C₄×C₄₇	C₂×D₄	1	2	yes	—
39	C₂²×C₄×C₄₇	C₂×Q₈	1	4	yes	—
40	C₂²×C₄×C₄₇	C₂×Q₈	1	2	yes	—
41	C₂²×C₄×C₄₇	#41	1	8	yes	—
42	C₂²×Dic₄₇	—	2	8	no	—
43	C₂²×Dic₄₇	—	2	16	no	—
44	C₂²×Dic₄₇	—	2	16	no	—
45	C₂²×Dic₄₇	—	2	8	no	—
46	C₂²×Dic₄₇	#41	1	16	yes	—
47	C₄²×C₄₇	—	2	16	no	—
48	C₄²×C₄₇	C₄⋊C₄	1	16	yes	—
49	C₄⋊Dic₄₇	—	2	32	no	—
50	C₄×Dic₄₇	—	4	16	no	—
51	C₂³×C₄₇	C₂³×C₄	1	1	yes	characteristic
52	C₂³×C₄₇	C₂²×D₄	1	1	yes	characteristic
53	C₂³×C₄₇	C₂²×Q₈	1	1	yes	characteristic
54	C₂³×C₄₇	C₂×#41	1	8	yes	—
55	C₂³×C₄₇	C₂×#41	1	4	yes	—
56	C₂×C₄×C₄₇	—	2	8	no	—
57	C₂×C₄×C₄₇	—	2	8	no	—
58	C₂×C₄×C₄₇	—	2	8	no	—
59	C₂×C₄×C₄₇	—	2	8	no	—
60	C₂×C₄×C₄₇	—	2	16	no	—
61	C₂×C₄×C₄₇	—	4	16	no	—
62	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	8	yes	—
63	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	4	yes	—
64	C₂×C₄×C₄₇	#117	1	8	yes	—
65	C₂×C₄×C₄₇	#124	1	8	yes	—
66	C₂×C₄×C₄₇	#125	1	8	yes	—
67	C₂×Dic₄₇	—	2	16	no	—
68	C₂×Dic₄₇	—	2	32	no	—
69	C₂×Dic₄₇	—	4	8	no	—
70	C₂×Dic₄₇	—	4	8	no	—
71	C₂×Dic₄₇	—	4	16	no	—
72	#13854	—	47	1	no	maximal, Sylow
73	C₂²×C₄₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
74	C₂²×C₄₇	C₂×#117	1	2	yes	—
75	C₂²×C₄₇	C₄×#41	1	4	yes	—
76	C₂²×C₄₇	#525	1	4	yes	—
77	C₂²×C₄₇	C₂×#124	1	2	yes	—
78	C₂²×C₄₇	C₂×#125	1	2	yes	—
79	C₂²×C₄₇	#529	1	16	yes	—
80	C₂²×C₄₇	#533	1	4	yes	—
81	C₄×C₄₇	—	2	8	no	—
82	C₄×C₄₇	—	4	8	no	—
83	C₄×C₄₇	#382	1	8	yes	—
84	Dic₄₇	—	4	16	no	—
85	C₂×C₄×C₄⋊C₄	—	47	2	no	—
86	C₂×#376	D₄₇	1	1	yes	characteristic
87	#1442	—	47	8	no	—
88	C₄×#95	—	47	4	no	—
89	C₂×C₄₇	C₂×#382	1	1	yes	characteristic
90	C₂×C₄₇	#2296	1	2	yes	—
91	C₂×C₄₇	#2298	1	4	yes	—
92	C₂×C₄₇	#2398	1	4	yes	—
93	C₂×C₄₇	#2402	1	4	yes	—
94	C₂²×C₄⋊C₄	—	47	4	no	—
95	C₂²×C₄²	—	47	4	no	—
96	C₂²×C₄²	C₂×D₄₇	1	2	yes	—
97	C₂²×C₄²	C₂×D₄₇	1	1	yes	characteristic
98	C₄×C₄⋊C₄	—	94	8	no	—
99	#376	Dic₄₇	1	8	yes	complemented
100	C₂×#95	—	47	16	no	—
101	C₂×#95	—	47	4	no	—
102	C₂×#95	C₂×D₄₇	1	4	yes	—
103	C₄₇	#13854	1	1	yes	characteristic, complemented, Sylow
104	C₂³×C₄	—	47	8	no	—
105	C₂³×C₄	C₂²×D₄₇	1	4	yes	—
106	C₂³×C₄	C₂²×D₄₇	1	2	yes	—
107	C₂³×C₄	C₂²×D₄₇	1	1	yes	characteristic
108	C₂×C₄⋊C₄	—	94	16	no	—
109	C₂×C₄⋊C₄	—	94	8	no	—
110	C₂×C₄²	—	2	8	no	—
111	C₂×C₄²	—	94	8	no	—
112	C₂×C₄²	—	94	16	no	—
113	C₂×C₄²	Dic₉₄	1	4	yes	—
114	C₂×C₄²	D₁₈₈	1	4	yes	—
115	C₂×C₄²	C₂×Dic₄₇	1	8	yes	—
116	C₂×C₄²	C₂×Dic₄₇	1	4	yes	—
117	#95	—	94	32	no	—
118	#95	—	94	8	no	—
119	#95	C₂×Dic₄₇	1	16	yes	—
120	C₂⁴	C₂³×D₄₇	1	1	yes	characteristic
121	C₂²×C₄	—	2	8	no	—
122	C₂²×C₄	—	2	16	no	—
123	C₂²×C₄	—	2	8	no	—
124	C₂²×C₄	—	47	16	no	—
125	C₂²×C₄	—	94	8	no	—
126	C₂²×C₄	—	94	16	no	—
127	C₂²×C₄	—	94	8	no	—
128	C₂²×C₄	—	94	16	no	—
129	C₂²×C₄	C₂×Dic₉₄	1	4	yes	—
130	C₂²×C₄	C₂×Dic₉₄	1	2	yes	—
131	C₂²×C₄	C₂×D₁₈₈	1	4	yes	—
132	C₂²×C₄	C₂×D₁₈₈	1	2	yes	—
133	C₂²×C₄	#10625350	1	8	yes	—
134	C₂²×C₄	C₂²×Dic₄₇	1	8	yes	—
135	C₂²×C₄	C₂²×Dic₄₇	1	4	yes	—
136	C₂²×C₄	C₂²×Dic₄₇	1	2	yes	—
137	C₄⋊C₄	—	94	32	no	—
138	C₄²	—	2	16	no	—
139	C₄²	—	188	16	no	—
140	C₄²	C₄⋊Dic₄₇	1	16	yes	—
141	C₂³	C₂³×Dic₄₇	1	1	yes	characteristic
142	C₂³	C₂²×D₁₈₈	1	1	yes	characteristic
143	C₂³	C₂²×Dic₉₄	1	1	yes	characteristic
144	C₂³	#41×D₄₇	1	8	yes	—
145	C₂³	C₂×#10625350	1	4	yes	—
146	C₂×C₄	—	2	8	no	—
147	C₂×C₄	—	2	8	no	—
148	C₂×C₄	—	2	8	no	—
149	C₂×C₄	—	2	8	no	—
150	C₂×C₄	—	2	16	no	—
151	C₂×C₄	—	4	16	no	—
152	C₂×C₄	—	94	16	no	—
153	C₂×C₄	—	94	32	no	—
154	C₂×C₄	—	188	8	no	—
155	C₂×C₄	—	188	8	no	—
156	C₂×C₄	—	188	16	no	—
157	C₂×C₄	C₂×C₄⋊Dic₄₇	1	8	yes	—
158	C₂×C₄	C₂×C₄⋊Dic₄₇	1	4	yes	—
159	C₂×C₄	#13093983	1	8	yes	—
160	C₂×C₄	#13094012	1	8	yes	—
161	C₂×C₄	#13094016	1	8	yes	—
162	C₂²	C₂²×C₄⋊Dic₄₇	1	1	yes	characteristic
163	C₂²	C₂×#13093983	1	2	yes	—
164	C₂²	#41×Dic₄₇	1	4	yes	—
165	C₂²	#15689635	1	4	yes	—
166	C₂²	C₂×#13094012	1	2	yes	—
167	C₂²	C₂×#13094016	1	2	yes	—
168	C₂²	#15689651	1	16	yes	—
169	C₂²	#15689676	1	4	yes	—
170	C₄	—	2	8	no	—
171	C₄	—	4	8	no	—
172	C₄	—	188	16	no	—
173	C₄	#15688989	1	8	yes	—
174	C₂	C₂×#15688989	1	1	yes	characteristic
175	C₂	#20245872	1	2	yes	—
176	C₂	#20245886	1	4	yes	—
177	C₂	#20246674	1	4	yes	—
178	C₂	#20246693	1	4	yes	—

	subgroup	G/H
center	C₂³×C₄	C₂²×D₄₇
commutator	C₂³×C₄₇	C₂³×C₄
Frattini	C₂⁴	C₂³×D₄₇
Fitting	C₂×C₄₇×#376	C₂
socle, prime radical, cosocle	C₂⁴×C₄₇	C₂⁴
hypercenter	C₂×#376	D₄₇
hypocenter	C₄₇	#13854
LCS	G ▷ C₂³×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂³×C₄ ◁ C₂×#376

Complements

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

N	H ≅ G/N	type
12. C₄₇×#376	C₄	semidirect
99. #376	Dic₄₇	semidirect
103. C₄₇	#13854	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄×C₄⋊Dic₄₇	2
#20241645	2
C₄×C₄₇⋊#95	2
C₂×C₄₇×#376	2
#13854	47