Subgroups — #13271945

Order 1792.

280 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	C₇×#1848	C₂	1	1	yes	maximal, characteristic
2	#11728439	C₂	1	2	yes	maximal
3	#11727533	C₂	1	1	yes	maximal, characteristic
4	C₂×#89258	C₂	1	1	yes	maximal, characteristic
5	#11725540	C₂	1	2	yes	maximal
6	C₂×C₇×#108	—	2	1	no	—
7	C₂×C₇×#108	C₂²	1	2	yes	—
8	C₇×#423	—	2	1	no	—
9	C₂×C₄×C₇×C₈	C₂²	1	1	yes	characteristic
10	C₂×#6393	C₂²	1	1	yes	characteristic
11	#89878	C₂²	1	1	yes	characteristic
12	#89873	—	2	1	no	—
13	C₂×#6386	C₂²	1	2	yes	—
14	#89318	—	2	1	no	—
15	#89289	—	2	4	no	—
16	#89258	C₄	1	2	yes	complemented
17	#89258	C₄	1	2	yes	complemented
18	#13188	—	7	1	no	maximal, Sylow
19	C₂×C₇×M₁₆	—	2	2	no	—
20	C₂²×C₇×C₈	D₄	1	1	yes	characteristic
21	C₂²×C₇×C₈	C₂³	1	1	yes	characteristic
22	C₂×C₄²×C₇	D₄	1	1	yes	characteristic
23	#6489	—	2	1	no	—
24	C₂×#1136	—	2	1	no	—
25	C₇×#108	—	2	2	no	—
26	C₇×#108	—	2	2	no	—
27	C₇×#108	—	4	1	no	—
28	C₇×#108	—	4	1	no	—
29	C₇×C₄⋊C₈	—	4	2	no	—
30	C₄×C₇×C₈	C₂×C₄	1	1	yes	characteristic
31	C₄×C₇×C₈	C₂×C₄	1	1	yes	characteristic
32	C₄×C₇×C₈	Q₈	1	1	yes	characteristic
33	C₄×C₇×C₈	Q₈	1	1	yes	characteristic
34	#6393	—	2	1	no	—
35	#6393	Q₈	1	2	yes	—
36	#6392	—	2	1	no	—
37	#6391	D₄	1	2	yes	—
38	#6389	—	4	1	no	—
39	C₈×Dic₇	—	4	1	no	—
40	#6386	—	2	2	no	—
41	#6386	C₂×C₄	1	4	yes	—
42	C₂×C₇⋊C₁₆	—	2	2	no	—
43	#2155	—	7	2	no	—
44	C₂×#364	—	7	1	no	—
45	#1848	D₇	1	1	yes	characteristic
46	#1772	—	7	1	no	—
47	#1388	—	7	2	no	—
48	C₂²×C₄×C₇	—	2	1	no	—
49	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
50	C₇×M₁₆	—	4	2	no	—
51	C₇×M₁₆	—	4	2	no	—
52	C₂×C₇×C₈	—	4	2	no	—
53	C₂×C₇×C₈	C₄⋊C₄	1	1	yes	characteristic
54	C₂×C₇×C₈	C₄⋊C₄	1	1	yes	characteristic
55	C₂×C₇×C₈	SD₁₆	1	1	yes	characteristic
56	C₂×C₇×C₈	SD₁₆	1	1	yes	characteristic
57	C₂×C₇×C₈	SD₁₆	1	1	yes	characteristic
58	C₂×C₇×C₈	SD₁₆	1	1	yes	characteristic
59	C₂×C₇×C₈	C₂²×C₄	1	1	yes	characteristic
60	C₂×C₇×C₈	C₂×D₄	1	1	yes	characteristic
61	C₂×C₇×C₈	C₂×Q₈	1	1	yes	characteristic
62	C₂×C₇×C₈	C₂×Q₈	1	1	yes	characteristic
63	C₂×C₇×C₈	#41	1	1	yes	characteristic
64	C₂×C₇×C₈	#41	1	1	yes	characteristic
65	C₄²×C₇	—	2	1	no	—
66	C₄²×C₇	C₄⋊C₄	1	1	yes	characteristic
67	C₄²×C₇	C₄⋊C₄	1	1	yes	characteristic
68	C₂²⋊Dic₇	—	4	1	no	—
69	C₄⋊Dic₇	—	2	2	no	—
70	C₄×Dic₇	—	4	1	no	—
71	#1136	—	2	2	no	—
72	#1136	—	4	1	no	—
73	C₂×C₇⋊C₈	—	4	1	no	—
74	C₇⋊C₁₆	—	2	4	no	—
75	C₂×M₃₂	—	7	2	no	—
76	C₂×#108	—	2	1	no	—
77	C₂×#108	—	7	1	no	—
78	C₂×#108	C₂×D₇	1	2	yes	—
79	#427	—	7	1	no	—
80	#423	—	2	1	no	—
81	#405	—	14	1	no	—
82	C₂×C₄×C₈	C₂×D₇	1	1	yes	characteristic
83	#365	—	14	4	no	—
84	#364	—	7	2	no	—
85	#364	—	7	2	no	—
86	#339	—	14	1	no	—
87	C₂³×C₇	#137	1	1	yes	characteristic
88	C₂×C₄×C₇	—	2	1	no	—
89	C₂×C₄×C₇	—	2	1	no	—
90	C₂×C₄×C₇	—	4	2	no	—
91	C₂×C₄×C₇	C₂×C₄⋊C₄	1	1	yes	characteristic
92	C₂×C₄×C₇	#121	1	1	yes	characteristic
93	C₂×C₄×C₇	C₂²⋊Q₈	1	1	yes	characteristic
94	C₂×C₄×C₇	C₂×SD₁₆	1	1	yes	characteristic
95	C₂×C₄×C₇	C₂×SD₁₆	1	1	yes	characteristic
96	C₂×C₄×C₇	AGL(1,ℤ/8)	1	1	yes	characteristic
97	C₂×Dic₇	—	4	2	no	—
98	C₇×C₈	—	8	2	no	—
99	C₇×C₈	#106	1	1	yes	characteristic
100	C₇×C₈	#106	1	1	yes	characteristic
101	C₇×C₈	#106	1	1	yes	characteristic
102	C₇×C₈	#106	1	1	yes	characteristic
103	C₇×C₈	C₄×D₄	1	1	yes	characteristic
104	C₇×C₈	C₄×Q₈	1	1	yes	characteristic
105	C₇×C₈	#125	1	1	yes	characteristic
106	C₇×C₈	C₄⋊Q₈	1	1	yes	characteristic
107	C₇⋊C₈	—	4	2	no	—
108	C₂×M₁₆	—	2	2	no	—
109	C₂×M₁₆	—	14	1	no	—
110	C₂²×C₈	C₂²⋊D₇	1	1	yes	characteristic
111	C₂²×C₈	C₂²×D₇	1	1	yes	characteristic
112	#117	—	14	1	no	—
113	C₂×C₄²	C₂²⋊D₇	1	1	yes	characteristic
114	M₃₂	—	7	4	no	—
115	M₃₂	—	14	2	no	—
116	C₂×C₁₆	—	14	2	no	—
117	#108	—	2	2	no	—
118	#108	—	2	2	no	—
119	#108	—	4	1	no	—
120	#108	—	4	1	no	—
121	#108	—	7	2	no	—
122	#108	—	14	1	no	—
123	#107	—	14	1	no	—
124	#106	—	7	2	no	—
125	C₄⋊C₈	—	4	2	no	—
126	#97	—	28	1	no	—
127	C₄×C₈	—	28	1	no	—
128	C₄×C₈	Dic₁₄	1	1	yes	characteristic
129	C₄×C₈	Dic₁₄	1	1	yes	characteristic
130	C₄×C₈	C₄×D₇	1	1	yes	characteristic
131	C₄×C₈	C₄×D₇	1	1	yes	characteristic
132	C₂²×C₇	—	4	1	no	—
133	C₂²×C₇	#428	1	1	yes	characteristic
134	C₂²×C₇	#461	1	1	yes	characteristic
135	C₂²×C₇	#475	1	1	yes	characteristic
136	C₄×C₇	—	2	2	no	—
137	C₄×C₇	—	4	1	no	—
138	C₄×C₇	#389	1	1	yes	characteristic
139	C₄×C₇	C₂×#106	1	1	yes	characteristic
140	C₄×C₇	#460	1	1	yes	characteristic
141	C₄×C₇	#476	1	1	yes	characteristic
142	Dic₇	—	8	2	no	—
143	C₂²×C₄	—	2	1	no	—
144	C₂²×C₄	C₂×C₂²⋊D₇	1	1	yes	characteristic
145	M₁₆	—	4	2	no	—
146	M₁₆	—	4	2	no	—
147	M₁₆	—	14	2	no	—
148	M₁₆	—	28	1	no	—
149	C₂×C₈	—	4	2	no	—
150	C₂×C₈	—	28	1	no	—
151	C₂×C₈	Dic₇⋊C₄	1	1	yes	characteristic
152	C₂×C₈	Dic₇⋊C₄	1	1	yes	characteristic
153	C₂×C₈	#1142	1	1	yes	characteristic
154	C₂×C₈	#1142	1	1	yes	characteristic
155	C₂×C₈	#1143	1	1	yes	characteristic
156	C₂×C₈	#1143	1	1	yes	characteristic
157	C₂×C₈	C₂×Dic₁₄	1	1	yes	characteristic
158	C₂×C₈	C₂×C₄×D₇	1	1	yes	characteristic
159	C₂×C₈	D₄×D₇	1	1	yes	characteristic
160	C₂×C₈	#1159	1	1	yes	characteristic
161	C₂×C₈	Q₈×D₇	1	1	yes	characteristic
162	C₂×C₈	#1161	1	1	yes	characteristic
163	C₄⋊C₄	—	14	2	no	—
164	C₂²⋊C₄	—	28	1	no	—
165	C₄²	—	2	1	no	—
166	C₄²	—	28	1	no	—
167	C₄²	Dic₇⋊C₄	1	1	yes	characteristic
168	C₄²	Dic₇⋊C₄	1	1	yes	characteristic
169	C₁₆	—	14	4	no	—
170	C₂×C₇	—	2	1	no	—
171	C₂×C₇	—	2	1	no	—
172	C₂×C₇	—	2	1	no	—
173	C₂×C₇	#1948	1	1	yes	characteristic
174	C₂³	#6505	1	1	yes	characteristic
175	C₂×C₄	—	2	1	no	—
176	C₂×C₄	—	2	1	no	—
177	C₂×C₄	—	4	2	no	—
178	C₂×C₄	—	28	2	no	—
179	C₂×C₄	C₂×Dic₇⋊C₄	1	1	yes	characteristic
180	C₂×C₄	#6495	1	1	yes	characteristic
181	C₂×C₄	C₂×#1142	1	1	yes	characteristic
182	C₂×C₄	#6501	1	1	yes	characteristic
183	C₂×C₄	C₂×#1143	1	1	yes	characteristic
184	C₂×C₄	#6509	1	1	yes	characteristic
185	C₈	—	8	2	no	—
186	C₈	—	28	2	no	—
187	C₈	#6382	1	1	yes	characteristic
188	C₈	#6382	1	1	yes	characteristic
189	C₈	#6382	1	1	yes	characteristic
190	C₈	#6382	1	1	yes	characteristic
191	C₈	#6450	1	1	yes	characteristic
192	C₈	#6451	1	1	yes	characteristic
193	C₈	#6453	1	1	yes	characteristic
194	C₈	#6456	1	1	yes	characteristic
195	C₇	#13188	1	1	yes	characteristic, complemented, Sylow
196	C₂²	—	4	1	no	—
197	C₂²	#89736	1	1	yes	characteristic
198	C₂²	#89855	1	1	yes	characteristic
199	C₂²	#89866	1	1	yes	characteristic
200	C₄	—	2	2	no	—
201	C₄	—	4	1	no	—
202	C₄	—	56	2	no	—
203	C₄	C₂×#6382	1	1	yes	characteristic
204	C₄	#89745	1	1	yes	characteristic
205	C₄	#89847	1	1	yes	characteristic
206	C₄	#89856	1	1	yes	characteristic
207	C₂	—	2	1	no	—
208	C₂	—	2	1	no	—
209	C₂	—	2	1	no	—
210	C₂	#11727322	1	1	yes	characteristic

	subgroup	G/H
center	C₄	#89856
commutator	C₂×C₇×C₈	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₇
Fitting	C₇×#1848	C₂
socle	C₂×C₇	#1948
prime radical	C₂³×C₇	#137
cosocle	C₂²×C₇×C₈	C₂³
hypercenter	#1848	D₇
hypocenter	C₇	#13188
LCS	G ▷ C₂×C₇×C₈ ▷ C₄×C₇ ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂×C₄×C₈ ◁ #1848

Complements

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

N	H ≅ G/N	type
16. #89258	C₄	semidirect
17. #89258	C₄	semidirect
195. C₇	#13188	semidirect

Maximal subgroups

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

subgroup	index
C₇×#1848	2
#11728439	2
#11727533	2
C₂×#89258	2
#11725540	2
#13188	7