Subgroups — #13265396

Order 1792.

262 conjugacy classes of subgroups.

The subgroup lattice has 190 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₇×C₈²	C₂	1	1	yes	maximal, characteristic
2	#11727498	C₂	1	2	yes	maximal
3	#11725246	C₂	1	2	yes	maximal, complemented
4	#11725245	C₂	1	2	yes	maximal, complemented
5	C₂×C₄×C₇×C₈	C₄	1	1	yes	characteristic
6	C₂×C₄×C₇×C₈	C₄	1	1	yes	characteristic
7	C₂×C₄×C₇×C₈	C₂²	1	1	yes	characteristic
8	C₇×C₈²	C₄	1	1	yes	characteristic
9	C₇×C₈²	C₄	1	1	yes	characteristic
10	C₇×C₈²	C₂²	1	1	yes	characteristic
11	C₇×C₈²	C₂²	1	1	yes	characteristic
12	C₂²⋊(C₇⋊C₁₆)	—	2	2	no	—
13	C₄⋊(C₇⋊C₁₆)	C₂²	1	4	yes	—
14	C₄×C₇⋊C₁₆	—	2	2	no	—
15	#12012	—	7	1	no	maximal, Sylow
16	C₂²×C₇×C₈	—	2	1	no	—
17	C₂²×C₇×C₈	C₈	1	1	yes	characteristic
18	C₂²×C₇×C₈	C₈	1	1	yes	characteristic
19	C₂²×C₇×C₈	D₄	1	1	yes	characteristic
20	C₂²×C₇×C₈	Q₈	1	1	yes	characteristic
21	C₂×C₄²×C₇	C₂×C₄	1	1	yes	characteristic
22	C₄×C₇×C₈	—	2	1	no	—
23	C₄×C₇×C₈	C₈	1	1	yes	characteristic
24	C₄×C₇×C₈	C₈	1	1	yes	characteristic
25	C₄×C₇×C₈	C₂×C₄	1	1	yes	characteristic
26	C₄×C₇×C₈	C₂×C₄	1	1	yes	characteristic
27	C₄×C₇×C₈	C₂×C₄	1	1	yes	characteristic
28	C₄×C₇×C₈	C₂×C₄	1	1	yes	characteristic
29	C₄×C₇×C₈	C₂×C₄	1	1	yes	characteristic
30	C₄×C₇×C₈	D₄	1	1	yes	characteristic
31	C₄×C₇×C₈	Q₈	1	1	yes	characteristic
32	C₄×C₇×C₈	C₂³	1	1	yes	characteristic
33	C₂×C₇⋊C₁₆	—	2	4	no	—
34	#2162	—	7	2	no	—
35	C₂×C₈²	D₇	1	1	yes	characteristic
36	#1370	—	7	2	no	—
37	#1367	—	7	2	no	—
38	C₂²×C₄×C₇	C₄⋊C₄	1	1	yes	characteristic
39	C₂²×C₄×C₇	C₂×C₈	1	1	yes	characteristic
40	C₂²×C₄×C₇	M₁₆	1	1	yes	characteristic
41	C₂×C₇×C₈	—	2	4	no	—
42	C₂×C₇×C₈	—	2	2	no	—
43	C₂×C₇×C₈	—	2	1	no	—
44	C₂×C₇×C₈	—	2	1	no	—
45	C₂×C₇×C₈	—	2	2	no	—
46	C₂×C₇×C₈	—	2	2	no	—
47	C₂×C₇×C₈	C₁₆	1	4	yes	complemented
48	C₂×C₇×C₈	C₄⋊C₄	1	1	yes	characteristic
49	C₂×C₇×C₈	C₄⋊C₄	1	1	yes	characteristic
50	C₂×C₇×C₈	C₂×C₈	1	1	yes	characteristic
51	C₂×C₇×C₈	C₂×C₈	1	1	yes	characteristic
52	C₂×C₇×C₈	C₂×C₈	1	1	yes	characteristic
53	C₂×C₇×C₈	C₂×C₈	1	1	yes	characteristic
54	C₂×C₇×C₈	C₂×D₄	1	1	yes	characteristic
55	C₂×C₇×C₈	C₂×Q₈	1	1	yes	characteristic
56	C₄²×C₇	C₄⋊C₄	1	1	yes	characteristic
57	C₄²×C₇	C₂×C₈	1	1	yes	characteristic
58	C₄²×C₇	M₁₆	1	1	yes	characteristic
59	C₄²×C₇	C₂²×C₄	1	1	yes	characteristic
60	C₇⋊C₁₆	—	4	4	no	—
61	C₂×C₄×C₈	Dic₇	1	1	yes	characteristic
62	C₂×C₄×C₈	Dic₇	1	1	yes	characteristic
63	C₂×C₄×C₈	C₂×D₇	1	1	yes	characteristic
64	C₄⋊C₁₆	—	7	4	no	—
65	C₂²⋊C₁₆	—	14	2	no	—
66	C₄×C₁₆	—	14	2	no	—
67	C₈²	Dic₇	1	1	yes	characteristic
68	C₈²	Dic₇	1	1	yes	characteristic
69	C₈²	C₂×D₇	1	1	yes	characteristic
70	C₈²	C₂×D₇	1	1	yes	characteristic
71	C₂³×C₇	C₄⋊C₈	1	1	yes	characteristic
72	C₂×C₄×C₇	—	2	2	no	—
73	C₂×C₄×C₇	—	2	1	no	—
74	C₂×C₄×C₇	—	2	1	no	—
75	C₂×C₄×C₇	C₄⋊C₈	1	1	yes	characteristic
76	C₂×C₄×C₇	C₄⋊C₈	1	1	yes	characteristic
77	C₂×C₄×C₇	C₄⋊C₈	1	1	yes	characteristic
78	C₂×C₄×C₇	C₂×C₁₆	1	2	yes	—
79	C₂×C₄×C₇	M₃₂	1	2	yes	—
80	C₂×C₄×C₇	C₂×C₄⋊C₄	1	1	yes	characteristic
81	C₂×C₄×C₇	C₂²×C₈	1	1	yes	characteristic
82	C₂×C₄×C₇	C₂×M₁₆	1	1	yes	characteristic
83	C₇×C₈	—	2	2	no	—
84	C₇×C₈	—	2	2	no	—
85	C₇×C₈	—	2	1	no	—
86	C₇×C₈	—	2	1	no	—
87	C₇×C₈	—	2	1	no	—
88	C₇×C₈	—	2	1	no	—
89	C₇×C₈	#108	1	2	yes	—
90	C₇×C₈	C₂×C₁₆	1	2	yes	—
91	C₇×C₈	C₂×C₁₆	1	2	yes	—
92	C₇×C₈	#135	1	2	yes	—
93	C₂²×C₈	—	2	1	no	—
94	C₂²×C₈	C₇⋊C₈	1	1	yes	characteristic
95	C₂²×C₈	C₇⋊C₈	1	1	yes	characteristic
96	C₂²×C₈	Dic₁₄	1	1	yes	characteristic
97	C₂²×C₈	D₂₈	1	1	yes	characteristic
98	C₂×C₄²	C₂×Dic₇	1	1	yes	characteristic
99	C₂×C₁₆	—	14	4	no	—
100	C₄×C₈	—	2	1	no	—
101	C₄×C₈	C₇⋊C₈	1	1	yes	characteristic
102	C₄×C₈	C₇⋊C₈	1	1	yes	characteristic
103	C₄×C₈	Dic₁₄	1	1	yes	characteristic
104	C₄×C₈	D₂₈	1	1	yes	characteristic
105	C₄×C₈	C₂×Dic₇	1	1	yes	characteristic
106	C₄×C₈	C₂×Dic₇	1	1	yes	characteristic
107	C₄×C₈	C₂×Dic₇	1	1	yes	characteristic
108	C₄×C₈	C₂×Dic₇	1	1	yes	characteristic
109	C₄×C₈	C₂×Dic₇	1	1	yes	characteristic
110	C₄×C₈	C₂²×D₇	1	1	yes	characteristic
111	C₂²×C₇	—	2	1	no	—
112	C₂²×C₇	—	2	1	no	—
113	C₂²×C₇	C₄⋊C₁₆	1	2	yes	—
114	C₂²×C₇	C₂×C₄⋊C₈	1	1	yes	characteristic
115	C₄×C₇	—	2	1	no	—
116	C₄×C₇	—	2	1	no	—
117	C₄×C₇	C₄⋊C₁₆	1	2	yes	—
118	C₄×C₇	#364	1	2	yes	—
119	C₄×C₇	#427	1	1	yes	characteristic
120	C₄×C₇	C₂×#108	1	1	yes	characteristic
121	C₄×C₇	C₂²×C₁₆	1	1	yes	characteristic
122	C₄×C₇	C₂×M₃₂	1	1	yes	characteristic
123	C₂²×C₄	C₂×C₇⋊C₈	1	1	yes	characteristic
124	C₂²×C₄	#1136	1	1	yes	characteristic
125	C₂²×C₄	C₄⋊Dic₇	1	1	yes	characteristic
126	C₂×C₈	—	2	4	no	—
127	C₂×C₈	—	2	1	no	—
128	C₂×C₈	—	2	1	no	—
129	C₂×C₈	—	2	2	no	—
130	C₂×C₈	—	2	2	no	—
131	C₂×C₈	—	2	2	no	—
132	C₂×C₈	C₇⋊C₁₆	1	4	yes	complemented
133	C₂×C₈	C₂×C₇⋊C₈	1	1	yes	characteristic
134	C₂×C₈	C₂×C₇⋊C₈	1	1	yes	characteristic
135	C₂×C₈	C₂×C₇⋊C₈	1	1	yes	characteristic
136	C₂×C₈	C₂×C₇⋊C₈	1	1	yes	characteristic
137	C₂×C₈	C₄⋊Dic₇	1	1	yes	characteristic
138	C₂×C₈	C₄⋊Dic₇	1	1	yes	characteristic
139	C₂×C₈	C₂×Dic₁₄	1	1	yes	characteristic
140	C₂×C₈	C₂×D₂₈	1	1	yes	characteristic
141	C₄²	C₂×C₇⋊C₈	1	1	yes	characteristic
142	C₄²	#1136	1	1	yes	characteristic
143	C₄²	C₄⋊Dic₇	1	1	yes	characteristic
144	C₄²	C₂²×Dic₇	1	1	yes	characteristic
145	C₁₆	—	28	4	no	—
146	C₂×C₇	—	2	2	no	—
147	C₂×C₇	#1564	1	1	yes	characteristic
148	C₂×C₇	C₂×C₄⋊C₁₆	1	1	yes	characteristic
149	C₂×C₇	C₂×#364	1	1	yes	characteristic
150	C₂³	C₄⋊(C₇⋊C₈)	1	1	yes	characteristic
151	C₂×C₄	—	2	1	no	—
152	C₂×C₄	—	2	1	no	—
153	C₂×C₄	—	2	2	no	—
154	C₂×C₄	C₄⋊(C₇⋊C₈)	1	1	yes	characteristic
155	C₂×C₄	C₄⋊(C₇⋊C₈)	1	1	yes	characteristic
156	C₂×C₄	C₄⋊(C₇⋊C₈)	1	1	yes	characteristic
157	C₂×C₄	C₂×C₇⋊C₁₆	1	2	yes	—
158	C₂×C₄	#6386	1	2	yes	—
159	C₂×C₄	C₂²×C₇⋊C₈	1	1	yes	characteristic
160	C₂×C₄	C₂×#1136	1	1	yes	characteristic
161	C₂×C₄	C₂×C₄⋊Dic₇	1	1	yes	characteristic
162	C₈	—	2	2	no	—
163	C₈	—	2	2	no	—
164	C₈	—	2	1	no	—
165	C₈	—	2	1	no	—
166	C₈	—	2	1	no	—
167	C₈	—	2	1	no	—
168	C₈	C₂×C₇⋊C₁₆	1	2	yes	—
169	C₈	C₂×C₇⋊C₁₆	1	2	yes	—
170	C₈	#6393	1	2	yes	—
171	C₈	#6467	1	2	yes	—
172	C₇	#12012	1	1	yes	characteristic, complemented, Sylow
173	C₂²	—	2	1	no	—
174	C₂²	—	2	1	no	—
175	C₂²	C₄⋊(C₇⋊C₁₆)	1	2	yes	—
176	C₂²	C₂×C₄⋊(C₇⋊C₈)	1	1	yes	characteristic
177	C₄	—	2	1	no	—
178	C₄	—	2	1	no	—
179	C₄	C₄⋊(C₇⋊C₁₆)	1	2	yes	—
180	C₄	#89258	1	2	yes	—
181	C₄	C₂²×C₇⋊C₁₆	1	1	yes	characteristic
182	C₄	C₂×#6386	1	1	yes	characteristic
183	C₄	#89878	1	1	yes	characteristic
184	C₄	C₂×#6393	1	1	yes	characteristic
185	C₂	—	2	2	no	—
186	C₂	#11727483	1	1	yes	characteristic
187	C₂	C₂×C₄⋊(C₇⋊C₁₆)	1	1	yes	characteristic
188	C₂	C₂×#89258	1	1	yes	characteristic

	subgroup	G/H
center	C₄×C₈	D₂₈
commutator	C₄×C₇	C₂²×C₁₆
Frattini	C₄×C₈	C₂²×D₇
Fitting	C₂×C₇×C₈²	C₂
socle	C₂²×C₇	C₂×C₄⋊C₈
prime radical	C₂³×C₇	C₄⋊C₈
cosocle	C₄×C₇×C₈	C₂³
hypercenter	C₂×C₈²	D₇
hypocenter	C₇	#12012
LCS	G ▷ C₄×C₇ ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₄×C₈ ◁ C₂×C₄×C₈ ◁ C₂×C₈²

Complements

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

N	H ≅ G/N	type
3. #11725246	C₂	semidirect
4. #11725245	C₂	semidirect
47. C₂×C₇×C₈	C₁₆	semidirect
132. C₂×C₈	C₇⋊C₁₆	semidirect
172. C₇	#12012	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₇×C₈²	2
#11727498	2
#11725246	2
#11725245	2
#12012	7