Subgroups — #13243255

Order 1792.

718 conjugacy classes of subgroups.

The subgroup lattice has 218 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₄×C₇×C₈	C₂	1	1	yes	maximal, characteristic
2	C₂×#89697	C₂	1	2	yes	maximal
3	#11725460	C₂	1	4	yes	maximal, complemented
4	C₂³×C₇×C₈	C₄	1	2	yes	—
5	C₂²×C₄²×C₇	C₂²	1	1	yes	characteristic
6	C₂×C₄×C₇×C₈	—	2	4	no	—
7	C₂×C₄×C₇×C₈	C₄	1	2	yes	—
8	C₂×C₄×C₇×C₈	C₂²	1	2	yes	—
9	C₂×C₂²⋊(C₇⋊C₈)	—	2	2	no	—
10	#89697	—	2	4	no	—
11	#89697	C₄	1	8	yes	—
12	C₂×C₄⋊(C₇⋊C₈)	C₂²	1	4	yes	—
13	C₂×C₄×C₇⋊C₈	—	2	2	no	—
14	#7872	—	7	1	no	maximal, Sylow
15	C₂³×C₄×C₇	C₂×C₄	1	1	yes	characteristic
16	C₂³×C₄×C₇	D₄	1	1	yes	characteristic
17	C₂³×C₄×C₇	Q₈	1	1	yes	characteristic
18	C₂²×C₇×C₈	—	2	4	no	—
19	C₂²×C₇×C₈	—	2	4	no	—
20	C₂²×C₇×C₈	C₈	1	8	yes	complemented
21	C₂²×C₇×C₈	C₂×C₄	1	4	yes	—
22	C₂×C₄²×C₇	C₂×C₄	1	4	yes	—
23	C₂×C₄²×C₇	C₂×C₄	1	1	yes	characteristic
24	C₂×C₄²×C₇	D₄	1	1	yes	characteristic
25	C₂×C₄²×C₇	D₄	1	4	yes	—
26	C₂×C₄²×C₇	Q₈	1	1	yes	characteristic
27	C₂×C₄²×C₇	C₂³	1	1	yes	characteristic
28	C₂²×C₇⋊C₈	—	2	4	no	—
29	C₄×C₇×C₈	—	2	4	no	—
30	C₄×C₇×C₈	—	2	4	no	—
31	C₂²⋊(C₇⋊C₈)	—	2	8	no	—
32	C₄⋊(C₇⋊C₈)	—	2	4	no	—
33	C₄⋊(C₇⋊C₈)	C₂×C₄	1	8	yes	—
34	C₄×C₇⋊C₈	—	2	8	no	—
35	C₂×#431	—	7	2	no	—
36	C₂²×C₄×C₈	D₇	1	1	yes	characteristic
37	#1277	—	7	4	no	—
38	C₂⁴×C₇	C₄⋊C₄	1	1	yes	characteristic
39	C₂²×C₄×C₇	—	2	4	no	—
40	C₂²×C₄×C₇	—	2	4	no	—
41	C₂²×C₄×C₇	C₄²	1	2	yes	—
42	C₂²×C₄×C₇	C₂²⋊C₄	1	4	yes	—
43	C₂²×C₄×C₇	C₂²⋊C₄	1	2	yes	—
44	C₂²×C₄×C₇	C₄⋊C₄	1	1	yes	characteristic
45	C₂²×C₄×C₇	C₄⋊C₄	1	4	yes	—
46	C₂²×C₄×C₇	C₄⋊C₄	1	1	yes	characteristic
47	C₂²×C₄×C₇	C₄⋊C₄	1	1	yes	characteristic
48	C₂²×C₄×C₇	C₂×C₈	1	4	yes	—
49	C₂²×C₄×C₇	M₁₆	1	4	yes	—
50	C₂²×C₄×C₇	C₂²×C₄	1	1	yes	characteristic
51	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
52	C₂²×C₄×C₇	C₂×Q₈	1	1	yes	characteristic
53	C₂×C₇×C₈	—	2	16	no	—
54	C₂×C₇×C₈	—	2	4	no	—
55	C₂×C₇×C₈	—	2	4	no	—
56	C₂×C₇×C₈	C₂×C₈	1	8	yes	—
57	C₄²×C₇	C₄²	1	2	yes	—
58	C₄²×C₇	C₂²⋊C₄	1	4	yes	—
59	C₄²×C₇	C₂²⋊C₄	1	2	yes	—
60	C₄²×C₇	C₄⋊C₄	1	4	yes	—
61	C₄²×C₇	C₂²×C₄	1	2	yes	—
62	C₄²×C₇	C₂×D₄	1	2	yes	—
63	C₂×C₇⋊C₈	—	2	8	no	—
64	C₂×C₇⋊C₈	—	4	8	no	—
65	C₂³×C₈	Dic₇	1	2	yes	—
66	C₂²×C₄²	C₂×D₇	1	1	yes	characteristic
67	#431	—	7	8	no	—
68	#431	—	14	4	no	—
69	C₂×C₄⋊C₈	—	7	4	no	—
70	C₂×C₂²⋊C₈	—	14	2	no	—
71	C₂×C₄×C₈	—	2	4	no	—
72	C₂×C₄×C₈	—	14	2	no	—
73	C₂×C₄×C₈	Dic₇	1	2	yes	—
74	C₂×C₄×C₈	C₂×D₇	1	2	yes	—
75	C₂³×C₇	—	2	2	no	—
76	C₂³×C₇	—	2	2	no	—
77	C₂³×C₇	#95	1	2	yes	—
78	C₂³×C₇	C₄⋊C₈	1	4	yes	—
79	C₂³×C₇	C₂×C₄⋊C₄	1	1	yes	characteristic
80	C₂×C₄×C₇	—	2	2	no	—
81	C₂×C₄×C₇	—	2	2	no	—
82	C₂×C₄×C₇	—	2	8	no	—
83	C₂×C₄×C₇	—	2	8	no	—
84	C₂×C₄×C₇	#95	1	2	yes	—
85	C₂×C₄×C₇	#95	1	2	yes	—
86	C₂×C₄×C₇	#95	1	2	yes	—
87	C₂×C₄×C₇	C₄×C₈	1	4	yes	—
88	C₂×C₄×C₇	#97	1	4	yes	—
89	C₂×C₄×C₇	C₂²⋊C₈	1	8	yes	—
90	C₂×C₄×C₇	C₄⋊C₈	1	4	yes	—
91	C₂×C₄×C₇	#108	1	4	yes	—
92	C₂×C₄×C₇	C₂×C₄²	1	1	yes	characteristic
93	C₂×C₄×C₇	C₂×C₂²⋊C₄	1	2	yes	—
94	C₂×C₄×C₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
95	C₂×C₄×C₇	C₂×C₄⋊C₄	1	2	yes	—
96	C₂×C₄×C₇	C₂²×C₈	1	2	yes	—
97	C₂×C₄×C₇	C₂×M₁₆	1	2	yes	—
98	C₂×C₄×C₇	#135	1	4	yes	—
99	C₇×C₈	—	2	8	no	—
100	C₇⋊C₈	—	4	8	no	—
101	C₂³×C₄	Dic₁₄	1	1	yes	characteristic
102	C₂³×C₄	D₂₈	1	1	yes	characteristic
103	C₂³×C₄	C₂×Dic₇	1	1	yes	characteristic
104	C₂²×C₈	—	2	4	no	—
105	C₂²×C₈	—	2	4	no	—
106	C₂²×C₈	—	14	4	no	—
107	C₂²×C₈	C₇⋊C₈	1	8	yes	complemented
108	C₂²×C₈	C₂×Dic₇	1	4	yes	—
109	C₂×C₄²	Dic₁₄	1	1	yes	characteristic
110	C₂×C₄²	C₄×D₇	1	4	yes	—
111	C₂×C₄²	D₂₈	1	1	yes	characteristic
112	C₂×C₄²	C₂×Dic₇	1	1	yes	characteristic
113	C₂×C₄²	C₂²⋊D₇	1	4	yes	—
114	C₂×C₄²	C₂²×D₇	1	1	yes	characteristic
115	C₄⋊C₈	—	7	8	no	—
116	C₄⋊C₈	—	14	4	no	—
117	C₂²⋊C₈	—	14	8	no	—
118	C₄×C₈	—	2	4	no	—
119	C₄×C₈	—	2	4	no	—
120	C₄×C₈	—	14	8	no	—
121	C₂²×C₇	—	2	8	no	—
122	C₂²×C₇	—	2	2	no	—
123	C₂²×C₇	—	2	2	no	—
124	C₂²×C₇	#336	1	4	yes	—
125	C₂²×C₇	C₂×#95	1	1	yes	characteristic
126	C₂²×C₇	C₂×C₄⋊C₈	1	2	yes	—
127	C₂²×C₇	#427	1	2	yes	—
128	C₂²×C₇	C₂×#108	1	2	yes	—
129	C₄×C₇	—	2	2	no	—
130	C₄×C₇	—	2	2	no	—
131	C₄×C₇	#336	1	4	yes	—
132	C₄×C₇	#341	1	4	yes	—
133	C₄×C₇	C₂×C₄×C₈	1	1	yes	characteristic
134	C₄×C₇	C₂×#97	1	1	yes	characteristic
135	C₄×C₇	C₂×C₂²⋊C₈	1	2	yes	—
136	C₄×C₇	#416	1	4	yes	—
137	C₂⁴	C₄⋊Dic₇	1	1	yes	characteristic
138	C₂²×C₄	—	2	4	no	—
139	C₂²×C₄	—	2	4	no	—
140	C₂²×C₄	C₂×C₇⋊C₈	1	4	yes	—
141	C₂²×C₄	#1136	1	4	yes	—
142	C₂²×C₄	C₄×Dic₇	1	2	yes	—
143	C₂²×C₄	Dic₇⋊C₄	1	4	yes	—
144	C₂²×C₄	C₄⋊Dic₇	1	1	yes	characteristic
145	C₂²×C₄	C₄⋊Dic₇	1	1	yes	characteristic
146	C₂²×C₄	C₄⋊Dic₇	1	1	yes	characteristic
147	C₂²×C₄	(C₂×D₇)⋊C₄	1	4	yes	—
148	C₂²×C₄	C₂²⋊Dic₇	1	2	yes	—
149	C₂²×C₄	C₂×Dic₁₄	1	1	yes	characteristic
150	C₂²×C₄	C₂×D₂₈	1	1	yes	characteristic
151	C₂²×C₄	C₂²×Dic₇	1	1	yes	characteristic
152	C₂×C₈	—	2	16	no	—
153	C₂×C₈	—	2	4	no	—
154	C₂×C₈	—	2	4	no	—
155	C₂×C₈	—	14	8	no	—
156	C₂×C₈	—	28	8	no	—
157	C₂×C₈	C₂×C₇⋊C₈	1	8	yes	—
158	C₄²	C₄×Dic₇	1	2	yes	—
159	C₄²	Dic₇⋊C₄	1	4	yes	—
160	C₄²	(C₂×D₇)⋊C₄	1	4	yes	—
161	C₄²	C₂²⋊Dic₇	1	2	yes	—
162	C₄²	C₂×C₄×D₇	1	2	yes	—
163	C₄²	C₂×C₂²⋊D₇	1	2	yes	—
164	C₂×C₇	—	2	4	no	—
165	C₂×C₇	#1564	1	4	yes	—
166	C₂×C₇	C₂×#336	1	1	yes	characteristic
167	C₂×C₇	#1735	1	1	yes	characteristic
168	C₂×C₇	C₂×#341	1	1	yes	characteristic
169	C₂³	—	2	2	no	—
170	C₂³	—	2	2	no	—
171	C₂³	C₄⋊(C₇⋊C₈)	1	4	yes	—
172	C₂³	C₇⋊#95	1	2	yes	—
173	C₂³	C₂×C₄⋊Dic₇	1	1	yes	characteristic
174	C₂×C₄	—	2	2	no	—
175	C₂×C₄	—	2	2	no	—
176	C₂×C₄	—	2	8	no	—
177	C₂×C₄	—	2	8	no	—
178	C₂×C₄	C₄×C₇⋊C₈	1	4	yes	—
179	C₂×C₄	#6377	1	4	yes	—
180	C₂×C₄	C₄⋊(C₇⋊C₈)	1	4	yes	—
181	C₂×C₄	#6393	1	4	yes	—
182	C₂×C₄	C₂²⋊(C₇⋊C₈)	1	8	yes	—
183	C₂×C₄	C₇⋊#95	1	2	yes	—
184	C₂×C₄	C₇⋊#95	1	2	yes	—
185	C₂×C₄	C₇⋊#95	1	2	yes	—
186	C₂×C₄	#6467	1	4	yes	—
187	C₂×C₄	C₂²×C₇⋊C₈	1	2	yes	—
188	C₂×C₄	C₂×#1136	1	2	yes	—
189	C₂×C₄	C₂×C₄×Dic₇	1	1	yes	characteristic
190	C₂×C₄	C₂×Dic₇⋊C₄	1	2	yes	—
191	C₂×C₄	C₂×(C₂×D₇)⋊C₄	1	2	yes	—
192	C₂×C₄	C₂×C₂²⋊Dic₇	1	1	yes	characteristic
193	C₈	—	2	8	no	—
194	C₈	—	28	8	no	—
195	C₇	#7872	1	1	yes	characteristic, complemented, Sylow
196	C₂²	—	2	2	no	—
197	C₂²	—	2	2	no	—
198	C₂²	—	2	8	no	—
199	C₂²	#89319	1	4	yes	—
200	C₂²	C₂×C₄⋊(C₇⋊C₈)	1	2	yes	—
201	C₂²	#89878	1	2	yes	—
202	C₂²	C₂×#6393	1	2	yes	—
203	C₂²	C₂×C₇⋊#95	1	1	yes	characteristic
204	C₄	—	2	2	no	—
205	C₄	—	2	2	no	—
206	C₄	#89319	1	4	yes	—
207	C₄	#89347	1	4	yes	—
208	C₄	C₂×C₄×C₇⋊C₈	1	1	yes	characteristic
209	C₄	C₂×#6377	1	1	yes	characteristic
210	C₄	#89885	1	4	yes	—
211	C₄	C₂×C₂²⋊(C₇⋊C₈)	1	2	yes	—
212	C₂	—	2	4	no	—
213	C₂	#11727483	1	4	yes	—
214	C₂	C₂×#89319	1	1	yes	characteristic
215	C₂	#11729723	1	1	yes	characteristic
216	C₂	C₂×#89347	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄²	D₂₈
commutator	C₄×C₇	C₂×C₄×C₈
Frattini	C₂×C₄²	C₂²×D₇
Fitting	C₂²×C₄×C₇×C₈	C₂
socle	C₂³×C₇	C₂×C₄⋊C₄
prime radical	C₂⁴×C₇	C₄⋊C₄
cosocle	C₂×C₄²×C₇	C₂³
hypercenter	C₂²×C₄×C₈	D₇
hypocenter	C₇	#7872
LCS	G ▷ C₄×C₇ ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₂×C₄² ◁ C₂²×C₄² ◁ C₂²×C₄×C₈

Complements

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

N	H ≅ G/N	type
3. #11725460	C₂	semidirect
20. C₂²×C₇×C₈	C₈	semidirect
107. C₂²×C₈	C₇⋊C₈	semidirect
195. C₇	#7872	semidirect

Maximal subgroups

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

subgroup	index
C₂²×C₄×C₇×C₈	2
C₂×#89697	2
#11725460	2
#7872	7