Subgroups — #13270239

Order 1792.

350 conjugacy classes of subgroups.

The subgroup lattice has 194 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₇×#1845	C₂	1	1	yes	maximal, characteristic
2	C₂×#89285	C₂	1	2	yes	maximal
3	C₂×#89284	C₂	1	2	yes	maximal
4	#11727520	C₂	1	1	yes	maximal, characteristic
5	C₂×#89256	C₂	1	1	yes	maximal, characteristic
6	C₂×C₇×#107	—	2	1	no	—
7	C₂×C₇×#107	C₂²	1	2	yes	—
8	C₂×C₄×C₇×C₈	C₂²	1	1	yes	characteristic
9	C₇×#384	—	2	1	no	—
10	C₂×#6392	C₂²	1	1	yes	characteristic
11	C₂×#6392	C₂²	1	1	yes	characteristic
12	C₂×#6391	—	2	1	no	—
13	C₂²×C₇⋊C₁₆	C₂²	1	2	yes	—
14	#89708	—	2	1	no	—
15	#89285	—	2	4	no	—
16	#89284	—	2	4	no	—
17	#89256	C₄	1	4	yes	complemented
18	#12908	—	7	1	no	maximal, Sylow
19	C₂²×C₇×C₈	D₄	1	1	yes	characteristic
20	C₂²×C₇×C₈	C₂³	1	1	yes	characteristic
21	C₂×C₇×C₄⋊C₄	—	2	2	no	—
22	C₂×C₇×C₄⋊C₄	—	4	2	no	—
23	C₂×C₄²×C₇	D₄	1	1	yes	characteristic
24	C₂×C₄⋊Dic₇	—	2	1	no	—
25	C₂×C₄⋊Dic₇	—	2	1	no	—
26	C₇×#107	—	2	2	no	—
27	C₇×#107	—	2	2	no	—
28	C₇×#107	—	4	2	no	—
29	C₄×C₇×C₈	C₂×C₄	1	2	yes	—
30	C₄×C₇×C₈	Q₈	1	2	yes	—
31	C₇⋊#95	—	4	2	no	—
32	#6392	—	2	2	no	—
33	#6392	—	2	2	no	—
34	#6391	—	4	2	no	—
35	C₂×C₇⋊C₁₆	C₂×C₄	1	4	yes	—
36	C₂×C₇⋊C₁₆	D₄	1	4	yes	—
37	C₂×C₇⋊C₁₆	Q₈	1	4	yes	—
38	C₂×#367	—	7	2	no	—
39	C₂×#366	—	7	2	no	—
40	C₂×#346	—	7	1	no	—
41	#1846	—	7	1	no	—
42	#1845	D₇	1	1	yes	characteristic
43	C₂²×C₄×C₇	—	4	2	no	—
44	C₂²×C₄×C₇	D₈	1	2	yes	—
45	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
46	C₂²×Dic₇	—	4	1	no	—
47	C₂²×Dic₇	—	4	1	no	—
48	C₂×C₇×C₈	C₄⋊C₄	1	2	yes	—
49	C₂×C₇×C₈	D₈	1	2	yes	—
50	C₂×C₇×C₈	Q₁₆	1	2	yes	—
51	C₂×C₇×C₈	C₂²×C₄	1	1	yes	characteristic
52	C₂×C₇×C₈	C₂×D₄	1	2	yes	—
53	C₂×C₇×C₈	C₂×Q₈	1	2	yes	—
54	C₂×C₇×C₈	C₂×Q₈	1	1	yes	characteristic
55	C₇×C₄⋊C₄	—	4	2	no	—
56	C₇×C₄⋊C₄	—	4	2	no	—
57	C₇×C₄⋊C₄	—	8	4	no	—
58	C₄²×C₇	C₄⋊C₄	1	2	yes	—
59	C₄²×C₇	Q₁₆	1	2	yes	—
60	C₄⋊Dic₇	—	4	1	no	—
61	C₄⋊Dic₇	—	4	1	no	—
62	C₄⋊Dic₇	—	4	1	no	—
63	C₄⋊Dic₇	—	4	1	no	—
64	C₇⋊C₁₆	C₄⋊C₄	1	8	yes	complemented
65	C₂²×C₁₆	—	7	2	no	—
66	C₂×#107	—	2	1	no	—
67	C₂×#107	—	7	1	no	—
68	C₂×#107	—	7	1	no	—
69	C₂×#107	C₂×D₇	1	2	yes	—
70	C₂×#106	—	14	1	no	—
71	C₂×C₄×C₈	C₂×D₇	1	1	yes	characteristic
72	#384	—	2	1	no	—
73	#382	—	14	1	no	—
74	#367	—	14	4	no	—
75	#366	—	14	4	no	—
76	#346	—	7	4	no	—
77	C₂³×C₇	C₂×D₈	1	1	yes	characteristic
78	C₂×C₄×C₇	—	2	2	no	—
79	C₂×C₄×C₇	—	2	2	no	—
80	C₂×C₄×C₇	—	8	2	no	—
81	C₂×C₄×C₇	—	8	4	no	—
82	C₂×C₄×C₇	#107	1	4	yes	—
83	C₂×C₄×C₇	C₂×C₄⋊C₄	1	1	yes	characteristic
84	C₂×C₄×C₇	#127	1	1	yes	characteristic
85	C₂×C₄×C₇	C₄⋊Q₈	1	1	yes	characteristic
86	C₂×C₄×C₇	C₂×D₈	1	1	yes	characteristic
87	C₂×C₄×C₇	C₂×Q₁₆	1	1	yes	characteristic
88	C₂×C₄×C₇	C₂×Q₁₆	1	1	yes	characteristic
89	C₂×Dic₇	—	8	1	no	—
90	C₂×Dic₇	—	8	1	no	—
91	C₂×Dic₇	—	8	2	no	—
92	C₂×Dic₇	—	8	2	no	—
93	C₇×C₈	#107	1	4	yes	—
94	C₇×C₈	C₂×C₄⋊C₄	1	2	yes	—
95	C₇×C₈	C₄⋊Q₈	1	2	yes	—
96	C₂²×C₈	C₂²⋊D₇	1	1	yes	characteristic
97	C₂²×C₈	C₂²×D₇	1	1	yes	characteristic
98	C₂×C₄⋊C₄	—	2	2	no	—
99	C₂×C₄⋊C₄	—	4	2	no	—
100	C₂×C₄⋊C₄	—	14	1	no	—
101	C₂×C₄⋊C₄	—	14	1	no	—
102	C₂×C₄²	C₂²⋊D₇	1	1	yes	characteristic
103	C₂×C₁₆	—	7	4	no	—
104	C₂×C₁₆	—	7	4	no	—
105	C₂×C₁₆	—	7	4	no	—
106	#107	—	2	2	no	—
107	#107	—	2	2	no	—
108	#107	—	4	2	no	—
109	#107	—	14	2	no	—
110	#107	—	14	2	no	—
111	#106	—	28	2	no	—
112	C₄×C₈	Dic₁₄	1	2	yes	—
113	C₄×C₈	C₄×D₇	1	2	yes	—
114	#95	—	28	2	no	—
115	C₂²×C₇	C₂×#107	1	1	yes	characteristic
116	C₂²×C₇	#493	1	1	yes	characteristic
117	C₂²×C₇	#500	1	1	yes	characteristic
118	C₂²×C₇	#509	1	2	yes	—
119	C₂²×C₇	#510	1	2	yes	—
120	C₄×C₇	—	2	4	no	—
121	C₄×C₇	—	16	2	no	—
122	C₄×C₇	#384	1	1	yes	characteristic
123	C₄×C₇	C₂×#107	1	1	yes	characteristic
124	C₄×C₇	#494	1	1	yes	characteristic
125	C₄×C₇	#500	1	1	yes	characteristic
126	Dic₇	—	16	1	no	—
127	Dic₇	—	16	1	no	—
128	C₂²×C₄	—	4	2	no	—
129	C₂²×C₄	—	28	1	no	—
130	C₂²×C₄	—	28	1	no	—
131	C₂²×C₄	#1141	1	2	yes	—
132	C₂²×C₄	C₂×C₂²⋊D₇	1	1	yes	characteristic
133	C₂×C₈	Dic₇⋊C₄	1	2	yes	—
134	C₂×C₈	#1141	1	2	yes	—
135	C₂×C₈	#1144	1	2	yes	—
136	C₂×C₈	C₂×Dic₁₄	1	1	yes	characteristic
137	C₂×C₈	C₂×C₄×D₇	1	1	yes	characteristic
138	C₂×C₈	D₄×D₇	1	2	yes	—
139	C₂×C₈	Q₈×D₇	1	2	yes	—
140	C₄⋊C₄	—	4	2	no	—
141	C₄⋊C₄	—	4	2	no	—
142	C₄⋊C₄	—	8	4	no	—
143	C₄⋊C₄	—	28	1	no	—
144	C₄⋊C₄	—	28	1	no	—
145	C₄⋊C₄	—	28	1	no	—
146	C₄⋊C₄	—	28	1	no	—
147	C₄²	Dic₇⋊C₄	1	2	yes	—
148	C₄²	#1144	1	2	yes	—
149	C₁₆	—	7	8	no	—
150	C₂×C₇	#1845	1	1	yes	characteristic
151	C₂×C₇	#2159	1	2	yes	—
152	C₂×C₇	#2250	1	1	yes	characteristic
153	C₂×C₇	#2251	1	1	yes	characteristic
154	C₂×C₇	#2255	1	2	yes	—
155	C₂³	C₂×#1141	1	1	yes	characteristic
156	C₂×C₄	—	2	2	no	—
157	C₂×C₄	—	2	2	no	—
158	C₂×C₄	—	8	4	no	—
159	C₂×C₄	—	8	2	no	—
160	C₂×C₄	—	56	2	no	—
161	C₂×C₄	—	56	2	no	—
162	C₂×C₄	—	56	1	no	—
163	C₂×C₄	—	56	1	no	—
164	C₂×C₄	#6381	1	4	yes	—
165	C₂×C₄	C₂×Dic₇⋊C₄	1	1	yes	characteristic
166	C₂×C₄	C₂×#1141	1	1	yes	characteristic
167	C₂×C₄	#6503	1	1	yes	characteristic
168	C₂×C₄	C₂×#1144	1	1	yes	characteristic
169	C₂×C₄	C₂×#1144	1	1	yes	characteristic
170	C₂×C₄	#6507	1	1	yes	characteristic
171	C₈	#6381	1	4	yes	—
172	C₈	#6451	1	2	yes	—
173	C₈	D₇×C₄⋊C₄	1	2	yes	—
174	C₇	#12908	1	1	yes	characteristic, complemented, Sylow
175	C₂²	C₂×#6381	1	1	yes	characteristic
176	C₂²	#89845	1	1	yes	characteristic
177	C₂²	#89850	1	1	yes	characteristic
178	C₂²	#89919	1	2	yes	—
179	C₂²	#89951	1	2	yes	—
180	C₄	—	2	4	no	—
181	C₄	—	16	2	no	—
182	C₄	—	112	1	no	—
183	C₄	—	112	1	no	—
184	C₄	C₂×#6381	1	1	yes	characteristic
185	C₄	#89748	1	1	yes	characteristic
186	C₄	#89850	1	1	yes	characteristic
187	C₄	#89864	1	1	yes	characteristic
188	C₂	#11727331	1	1	yes	characteristic
189	C₂	#11728350	1	2	yes	—
190	C₂	#11729046	1	1	yes	characteristic
191	C₂	#11729068	1	1	yes	characteristic
192	C₂	#11729140	1	2	yes	—

	subgroup	G/H
center	C₂³	C₂×#1141
commutator	C₂×C₇×C₈	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₇
Fitting	C₇×#1845	C₂
socle, prime radical	C₂³×C₇	C₂×D₈
cosocle	C₂²×C₇×C₈	C₂³
hypercenter	#1845	D₇
hypocenter	C₇	#12908
LCS	G ▷ C₂×C₇×C₈ ▷ C₄×C₇ ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ #1845

Complements

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

N	H ≅ G/N	type
17. #89256	C₄	semidirect
64. C₇⋊C₁₆	C₄⋊C₄	semidirect
174. C₇	#12908	semidirect

Maximal subgroups

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

subgroup	index
C₇×#1845	2
C₂×#89285	2
C₂×#89284	2
#11727520	2
C₂×#89256	2
#12908	7