Subgroups — #13265398

Order 1792.

250 conjugacy classes of subgroups.

The subgroup lattice has 202 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₇×#2163	C₂	1	1	yes	maximal, characteristic
2	#11727499	C₂	1	1	yes	maximal, characteristic
3	C₂×C₈×C₇⋊C₈	C₂	1	1	yes	maximal, characteristic
4	#11725361	C₂	1	2	yes	maximal, complemented
5	#11725360	C₂	1	2	yes	maximal, complemented
6	C₂×C₄×C₇×C₈	C₂²	1	1	yes	characteristic
7	C₂×C₄×C₇⋊C₈	C₄	1	1	yes	characteristic
8	C₂×C₄×C₇⋊C₈	C₄	1	1	yes	characteristic
9	C₇×C₄⋊C₁₆	C₂²	1	2	yes	—
10	C₇×C₂²⋊C₁₆	—	2	1	no	—
11	C₇×#346	—	2	1	no	—
12	C₂²⋊(C₇⋊C₁₆)	—	2	1	no	—
13	C₄⋊(C₇⋊C₁₆)	C₂²	1	2	yes	—
14	#89256	—	2	1	no	—
15	C₈×C₇⋊C₈	C₄	1	1	yes	characteristic
16	C₈×C₇⋊C₈	C₄	1	1	yes	characteristic
17	C₈×C₇⋊C₈	C₂²	1	1	yes	characteristic
18	C₈×C₇⋊C₈	C₂²	1	1	yes	characteristic
19	#12013	—	7	1	no	maximal, Sylow
20	C₂²×C₇×C₈	D₄	1	1	yes	characteristic
21	C₂²×C₇×C₈	Q₈	1	1	yes	characteristic
22	C₂×C₄²×C₇	C₂×C₄	1	1	yes	characteristic
23	C₂²×C₇⋊C₈	—	2	1	no	—
24	C₂²×C₇⋊C₈	C₈	1	1	yes	characteristic
25	C₂²×C₇⋊C₈	C₈	1	1	yes	characteristic
26	C₂×C₇×C₁₆	—	2	2	no	—
27	C₄×C₇×C₈	C₂×C₄	1	1	yes	characteristic
28	C₄×C₇×C₈	D₄	1	1	yes	characteristic
29	C₄×C₇×C₈	Q₈	1	1	yes	characteristic
30	C₄×C₇×C₈	C₂³	1	1	yes	characteristic
31	C₂×C₇⋊C₁₆	—	2	2	no	—
32	C₄×C₇⋊C₈	—	2	1	no	—
33	C₄×C₇⋊C₈	C₈	1	1	yes	characteristic
34	C₄×C₇⋊C₈	C₈	1	1	yes	characteristic
35	C₄×C₇⋊C₈	C₂×C₄	1	1	yes	characteristic
36	C₄×C₇⋊C₈	C₂×C₄	1	1	yes	characteristic
37	C₄×C₇⋊C₈	C₂×C₄	1	1	yes	characteristic
38	C₄×C₇⋊C₈	C₂×C₄	1	1	yes	characteristic
39	#2163	—	7	1	no	—
40	#2163	D₇	1	1	yes	characteristic
41	C₂×C₈²	—	7	1	no	—
42	#1370	—	7	2	no	—
43	#1367	—	7	2	no	—
44	C₂²×C₄×C₇	C₄⋊C₄	1	1	yes	characteristic
45	C₂²×C₄×C₇	C₂×C₈	1	1	yes	characteristic
46	C₂²×C₄×C₇	M₁₆	1	1	yes	characteristic
47	C₂×C₇×C₈	—	2	2	no	—
48	C₂×C₇×C₈	—	2	2	no	—
49	C₂×C₇×C₈	C₄⋊C₄	1	1	yes	characteristic
50	C₂×C₇×C₈	C₄⋊C₄	1	1	yes	characteristic
51	C₂×C₇×C₈	C₂×D₄	1	1	yes	characteristic
52	C₂×C₇×C₈	C₂×Q₈	1	1	yes	characteristic
53	C₄²×C₇	C₄⋊C₄	1	1	yes	characteristic
54	C₄²×C₇	C₂×C₈	1	1	yes	characteristic
55	C₄²×C₇	M₁₆	1	1	yes	characteristic
56	C₄²×C₇	C₂²×C₄	1	1	yes	characteristic
57	C₂×C₇⋊C₈	—	2	4	no	—
58	C₂×C₇⋊C₈	—	2	1	no	—
59	C₂×C₇⋊C₈	—	2	1	no	—
60	C₂×C₇⋊C₈	—	2	2	no	—
61	C₂×C₇⋊C₈	—	2	2	no	—
62	C₂×C₇⋊C₈	C₂×C₈	1	1	yes	characteristic
63	C₂×C₇⋊C₈	C₂×C₈	1	1	yes	characteristic
64	C₂×C₇⋊C₈	C₂×C₈	1	1	yes	characteristic
65	C₂×C₇⋊C₈	C₂×C₈	1	1	yes	characteristic
66	C₇×C₁₆	—	4	2	no	—
67	C₇⋊C₁₆	—	4	2	no	—
68	C₂×C₄×C₈	—	7	1	no	—
69	C₂×C₄×C₈	—	7	1	no	—
70	C₂×C₄×C₈	C₂×D₇	1	1	yes	characteristic
71	C₄⋊C₁₆	—	7	2	no	—
72	C₄⋊C₁₆	C₂×D₇	1	2	yes	—
73	C₂²⋊C₁₆	—	2	1	no	—
74	C₂²⋊C₁₆	—	14	1	no	—
75	#346	—	2	1	no	—
76	#346	—	14	1	no	—
77	C₈²	—	7	1	no	—
78	C₈²	—	7	1	no	—
79	C₈²	—	7	1	no	—
80	C₈²	—	7	1	no	—
81	C₂³×C₇	C₄⋊C₈	1	1	yes	characteristic
82	C₂×C₄×C₇	—	2	2	no	—
83	C₂×C₄×C₇	—	2	1	no	—
84	C₂×C₄×C₇	—	2	1	no	—
85	C₂×C₄×C₇	—	2	1	no	—
86	C₂×C₄×C₇	—	2	1	no	—
87	C₂×C₄×C₇	C₄⋊C₈	1	1	yes	characteristic
88	C₂×C₄×C₇	C₄⋊C₈	1	1	yes	characteristic
89	C₂×C₄×C₇	C₄⋊C₈	1	1	yes	characteristic
90	C₂×C₄×C₇	C₂×C₄⋊C₄	1	1	yes	characteristic
91	C₂×C₄×C₇	C₂²×C₈	1	1	yes	characteristic
92	C₂×C₄×C₇	C₂×M₁₆	1	1	yes	characteristic
93	C₇×C₈	—	2	1	no	—
94	C₇×C₈	—	2	1	no	—
95	C₇×C₈	#108	1	2	yes	—
96	C₇×C₈	#135	1	2	yes	—
97	C₇⋊C₈	—	2	2	no	—
98	C₇⋊C₈	—	2	2	no	—
99	C₇⋊C₈	—	2	1	no	—
100	C₇⋊C₈	—	2	1	no	—
101	C₇⋊C₈	M₃₂	1	2	yes	—
102	C₇⋊C₈	M₃₂	1	2	yes	—
103	C₂²×C₈	—	7	1	no	—
104	C₂²×C₈	—	7	1	no	—
105	C₂²×C₈	—	14	1	no	—
106	C₂²×C₈	Dic₁₄	1	1	yes	characteristic
107	C₂²×C₈	C₂²⋊D₇	1	1	yes	characteristic
108	C₂×C₄²	C₄×D₇	1	1	yes	characteristic
109	C₂×C₁₆	—	2	2	no	—
110	C₂×C₁₆	—	14	2	no	—
111	C₄×C₈	—	7	1	no	—
112	C₄×C₈	—	7	1	no	—
113	C₄×C₈	—	7	1	no	—
114	C₄×C₈	—	7	1	no	—
115	C₄×C₈	—	7	1	no	—
116	C₄×C₈	—	7	1	no	—
117	C₄×C₈	—	14	1	no	—
118	C₄×C₈	Dic₁₄	1	1	yes	characteristic
119	C₄×C₈	C₄×D₇	1	1	yes	characteristic
120	C₄×C₈	C₂²⋊D₇	1	1	yes	characteristic
121	C₄×C₈	C₂²×D₇	1	1	yes	characteristic
122	C₂²×C₇	—	2	1	no	—
123	C₂²×C₇	—	2	1	no	—
124	C₂²×C₇	#364	1	2	yes	—
125	C₂²×C₇	C₂×C₄⋊C₈	1	1	yes	characteristic
126	C₄×C₇	—	2	1	no	—
127	C₄×C₇	—	2	1	no	—
128	C₄×C₇	—	2	1	no	—
129	C₄×C₇	—	2	1	no	—
130	C₄×C₇	#427	1	1	yes	characteristic
131	C₄×C₇	C₂×#108	1	1	yes	characteristic
132	C₄×C₇	C₂×M₃₂	1	1	yes	characteristic
133	C₄×C₇	C₂×M₃₂	1	1	yes	characteristic
134	C₂²×C₄	C₈×D₇	1	1	yes	characteristic
135	C₂²×C₄	#1131	1	1	yes	characteristic
136	C₂²×C₄	Dic₇⋊C₄	1	1	yes	characteristic
137	C₂×C₈	—	2	2	no	—
138	C₂×C₈	—	2	2	no	—
139	C₂×C₈	—	7	1	no	—
140	C₂×C₈	—	7	1	no	—
141	C₂×C₈	—	7	1	no	—
142	C₂×C₈	—	7	1	no	—
143	C₂×C₈	—	14	2	no	—
144	C₂×C₈	—	14	2	no	—
145	C₂×C₈	—	14	4	no	—
146	C₂×C₈	—	14	1	no	—
147	C₂×C₈	—	14	1	no	—
148	C₂×C₈	Dic₇⋊C₄	1	1	yes	characteristic
149	C₂×C₈	Dic₇⋊C₄	1	1	yes	characteristic
150	C₂×C₈	C₂×Dic₁₄	1	1	yes	characteristic
151	C₂×C₈	C₂×C₂²⋊D₇	1	1	yes	characteristic
152	C₄²	C₈×D₇	1	1	yes	characteristic
153	C₄²	#1131	1	1	yes	characteristic
154	C₄²	Dic₇⋊C₄	1	1	yes	characteristic
155	C₄²	C₂×C₄×D₇	1	1	yes	characteristic
156	C₁₆	—	4	2	no	—
157	C₁₆	—	28	2	no	—
158	C₂×C₇	—	2	2	no	—
159	C₂×C₇	#1564	1	1	yes	characteristic
160	C₂×C₇	#2150	1	1	yes	characteristic
161	C₂×C₇	C₂×#364	1	1	yes	characteristic
162	C₂³	Dic₇⋊C₈	1	1	yes	characteristic
163	C₂×C₄	—	2	2	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	Dic₇⋊C₈	1	1	yes	characteristic
169	C₂×C₄	Dic₇⋊C₈	1	1	yes	characteristic
170	C₂×C₄	Dic₇⋊C₈	1	1	yes	characteristic
171	C₂×C₄	C₂×C₈×D₇	1	1	yes	characteristic
172	C₂×C₄	C₂×#1131	1	1	yes	characteristic
173	C₂×C₄	C₂×Dic₇⋊C₄	1	1	yes	characteristic
174	C₈	—	2	1	no	—
175	C₈	—	2	1	no	—
176	C₈	—	7	2	no	—
177	C₈	—	7	2	no	—
178	C₈	—	14	2	no	—
179	C₈	—	14	2	no	—
180	C₈	—	14	1	no	—
181	C₈	—	14	1	no	—
182	C₈	#6396	1	2	yes	—
183	C₈	#6513	1	2	yes	—
184	C₇	#12013	1	1	yes	characteristic, complemented, Sylow
185	C₂²	—	2	1	no	—
186	C₂²	—	2	1	no	—
187	C₂²	#89306	1	2	yes	—
188	C₂²	C₂×Dic₇⋊C₈	1	1	yes	characteristic
189	C₄	—	2	1	no	—
190	C₄	—	2	1	no	—
191	C₄	—	2	1	no	—
192	C₄	—	2	1	no	—
193	C₄	D₇×M₃₂	1	1	yes	characteristic
194	C₄	D₇×M₃₂	1	1	yes	characteristic
195	C₄	#89770	1	1	yes	characteristic
196	C₄	C₂×#6396	1	1	yes	characteristic
197	C₂	—	2	2	no	—
198	C₂	#11728499	1	1	yes	characteristic
199	C₂	#11729174	1	1	yes	characteristic
200	C₂	C₂×#89306	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₈	C₂×C₂²⋊D₇
commutator	C₂×C₄×C₇	C₂²×C₈
Frattini	C₄×C₈	C₂²×D₇
Fitting	C₇×#2163	C₂
socle	C₂²×C₇	C₂×C₄⋊C₈
prime radical	C₂³×C₇	C₄⋊C₈
cosocle	C₄×C₇×C₈	C₂³
hypercenter	#2163	D₇
hypocenter	C₇	#12013
LCS	G ▷ C₂×C₄×C₇ ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₂×C₈ ◁ C₂×C₄×C₈ ◁ #2163

Complements

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

N	H ≅ G/N	type
4. #11725361	C₂	semidirect
5. #11725360	C₂	semidirect
184. C₇	#12013	semidirect

Maximal subgroups

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

subgroup	index
C₇×#2163	2
#11727499	2
C₂×C₈×C₇⋊C₈	2
#11725361	2
#11725360	2
#12013	7