Subgroups — #13271836

Order 1792.

318 conjugacy classes of subgroups.

The subgroup lattice has 198 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₇×#346	C₂	1	1	yes	maximal, characteristic
2	C₂×#89300	C₂	1	1	yes	maximal, characteristic
3	C₂×#89299	C₂	1	1	yes	maximal, characteristic
4	#11727524	C₂	1	2	yes	maximal
5	#11725612	C₂	1	2	yes	maximal
6	C₂²×C₇×C₁₆	C₂²	1	1	yes	characteristic
7	C₂²×C₇×C₁₆	C₂²	1	1	yes	characteristic
8	C₂×C₄×C₇×C₈	C₂²	1	1	yes	characteristic
9	C₂×#6392	C₂²	1	2	yes	—
10	C₂×#6392	C₂²	1	2	yes	—
11	#89700	—	2	2	no	—
12	C₇×#346	—	2	2	no	—
13	#89346	—	2	2	no	—
14	#89300	C₄	1	4	yes	complemented
15	#89299	—	2	2	no	—
16	#13167	—	7	1	no	maximal, Sylow
17	C₂²×C₇×C₈	D₄	1	1	yes	characteristic
18	C₂²×C₇×C₈	C₂³	1	1	yes	characteristic
19	C₂×C₄²×C₇	D₄	1	1	yes	characteristic
20	C₂×C₄⋊Dic₇	—	2	2	no	—
21	C₂×C₄⋊Dic₇	—	2	2	no	—
22	C₂×Dic₇⋊C₄	—	4	2	no	—
23	C₂×C₇×C₁₆	—	2	1	no	—
24	C₂×C₇×C₁₆	—	2	1	no	—
25	C₂×C₇×C₁₆	C₂×C₄	1	2	yes	—
26	C₂×C₇×C₁₆	D₄	1	2	yes	—
27	C₂×C₇×C₁₆	Q₈	1	2	yes	—
28	C₂×C₇×C₁₆	Q₈	1	2	yes	—
29	C₄×C₇×C₈	—	2	1	no	—
30	C₄×C₇×C₈	—	2	1	no	—
31	C₇⋊#95	—	4	2	no	—
32	#6392	—	2	2	no	—
33	#6392	—	2	2	no	—
34	#6392	—	2	2	no	—
35	#6392	C₂×C₄	1	4	yes	—
36	C₂×#367	—	7	1	no	—
37	C₂×#366	—	7	1	no	—
38	C₂×#346	D₇	1	1	yes	characteristic
39	#1942	—	7	2	no	—
40	#1380	—	7	2	no	—
41	C₂²×C₄×C₇	D₈	1	1	yes	characteristic
42	C₂²×C₄×C₇	D₈	1	1	yes	characteristic
43	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
44	C₂²×Dic₇	—	4	2	no	—
45	C₂²×Dic₇	—	4	2	no	—
46	C₂×C₇×C₈	—	2	1	no	—
47	C₂×C₇×C₈	—	2	1	no	—
48	C₂×C₇×C₈	Q₁₆	1	2	yes	—
49	C₂×C₇×C₈	C₂²×C₄	1	1	yes	characteristic
50	C₂×C₇×C₈	C₂×D₄	1	1	yes	characteristic
51	C₂×C₇×C₈	C₂×Q₈	1	1	yes	characteristic
52	C₂×C₇×C₈	C₂×Q₈	1	1	yes	characteristic
53	C₂×C₇×C₈	#41	1	1	yes	characteristic
54	C₂×C₇×C₈	#41	1	1	yes	characteristic
55	C₄²×C₇	—	2	1	no	—
56	C₄²×C₇	—	2	1	no	—
57	C₄⋊Dic₇	—	2	4	no	—
58	C₄⋊Dic₇	—	4	2	no	—
59	C₄⋊Dic₇	—	4	2	no	—
60	C₄⋊Dic₇	—	4	2	no	—
61	Dic₇⋊C₄	—	8	4	no	—
62	C₇×C₁₆	—	2	2	no	—
63	C₇×C₁₆	C₄⋊C₄	1	4	yes	complemented
64	C₂²×C₁₆	C₂×D₇	1	1	yes	characteristic
65	C₂²×C₁₆	C₂×D₇	1	1	yes	characteristic
66	C₂×#107	—	7	2	no	—
67	C₂×#107	—	7	2	no	—
68	C₂×C₄×C₈	C₂×D₇	1	1	yes	characteristic
69	#389	—	14	2	no	—
70	#367	—	7	4	no	—
71	#366	—	14	2	no	—
72	#346	—	2	2	no	—
73	#340	—	14	2	no	—
74	C₂³×C₇	C₂×D₈	1	1	yes	characteristic
75	C₂×C₄×C₇	—	2	1	no	—
76	C₂×C₄×C₇	—	2	1	no	—
77	C₂×C₄×C₇	—	2	2	no	—
78	C₂×C₄×C₇	—	2	2	no	—
79	C₂×C₄×C₇	C₄×D₄	1	1	yes	characteristic
80	C₂×C₄×C₇	#121	1	1	yes	characteristic
81	C₂×C₄×C₇	C₄⋊Q₈	1	1	yes	characteristic
82	C₂×C₄×C₇	C₂×Q₁₆	1	1	yes	characteristic
83	C₂×C₄×C₇	#135	1	1	yes	characteristic
84	C₂×C₄×C₇	#135	1	1	yes	characteristic
85	C₂×Dic₇	—	4	4	no	—
86	C₂×Dic₇	—	8	2	no	—
87	C₂×Dic₇	—	8	2	no	—
88	C₂×Dic₇	—	8	2	no	—
89	C₂×Dic₇	—	8	2	no	—
90	C₂×Dic₇	—	8	2	no	—
91	C₇×C₈	—	2	2	no	—
92	C₇×C₈	C₂×C₄⋊C₄	1	1	yes	characteristic
93	C₇×C₈	C₄×Q₈	1	1	yes	characteristic
94	C₇×C₈	C₂²⋊Q₈	1	1	yes	characteristic
95	C₇×C₈	#125	1	1	yes	characteristic
96	C₂²×C₈	D₂₈	1	1	yes	characteristic
97	C₂²×C₈	C₂²×D₇	1	1	yes	characteristic
98	C₂×C₄⋊C₄	—	14	2	no	—
99	C₂×C₄⋊C₄	—	14	2	no	—
100	C₂×C₄⋊C₄	—	28	2	no	—
101	C₂×C₄²	D₂₈	1	1	yes	characteristic
102	C₂×C₁₆	—	2	1	no	—
103	C₂×C₁₆	—	2	1	no	—
104	C₂×C₁₆	Dic₁₄	1	2	yes	—
105	C₂×C₁₆	Dic₁₄	1	2	yes	—
106	C₂×C₁₆	C₄×D₇	1	2	yes	—
107	C₂×C₁₆	C₂²⋊D₇	1	2	yes	—
108	#107	—	7	4	no	—
109	#107	—	14	2	no	—
110	#107	—	14	2	no	—
111	#107	—	14	2	no	—
112	C₄×C₈	—	2	1	no	—
113	C₄×C₈	—	2	1	no	—
114	#95	—	28	2	no	—
115	C₂²×C₇	C₄×D₈	1	1	yes	characteristic
116	C₂²×C₇	#466	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	—	4	1	no	—
121	C₄×C₇	—	4	1	no	—
122	C₄×C₇	#389	1	1	yes	characteristic
123	C₄×C₇	C₄×Q₁₆	1	1	yes	characteristic
124	C₄×C₇	#467	1	1	yes	characteristic
125	C₄×C₇	#499	1	1	yes	characteristic
126	Dic₇	—	8	4	no	—
127	Dic₇	—	16	2	no	—
128	C₂²×C₄	—	28	2	no	—
129	C₂²×C₄	—	28	2	no	—
130	C₂²×C₄	D₅₆	1	1	yes	characteristic
131	C₂²×C₄	D₅₆	1	1	yes	characteristic
132	C₂²×C₄	C₂×D₂₈	1	1	yes	characteristic
133	C₂×C₈	—	2	1	no	—
134	C₂×C₈	—	2	1	no	—
135	C₂×C₈	Dic₂₈	1	2	yes	—
136	C₂×C₈	C₂×Dic₁₄	1	1	yes	characteristic
137	C₂×C₈	C₂×Dic₁₄	1	1	yes	characteristic
138	C₂×C₈	C₂×C₄×D₇	1	1	yes	characteristic
139	C₂×C₈	#1157	1	1	yes	characteristic
140	C₂×C₈	#1157	1	1	yes	characteristic
141	C₂×C₈	C₂×C₂²⋊D₇	1	1	yes	characteristic
142	C₄⋊C₄	—	14	4	no	—
143	C₄⋊C₄	—	28	2	no	—
144	C₄⋊C₄	—	28	2	no	—
145	C₄⋊C₄	—	28	2	no	—
146	C₄⋊C₄	—	56	4	no	—
147	C₄²	—	2	1	no	—
148	C₄²	—	2	1	no	—
149	C₁₆	—	2	2	no	—
150	C₁₆	Dic₇⋊C₄	1	4	yes	complemented
151	C₂×C₇	#1942	1	1	yes	characteristic
152	C₂×C₇	#2175	1	2	yes	—
153	C₂×C₇	#2218	1	1	yes	characteristic
154	C₂×C₇	#2219	1	1	yes	characteristic
155	C₂×C₇	#2255	1	2	yes	—
156	C₂³	C₂×D₅₆	1	1	yes	characteristic
157	C₂×C₄	—	2	1	no	—
158	C₂×C₄	—	2	1	no	—
159	C₂×C₄	—	2	2	no	—
160	C₂×C₄	—	2	2	no	—
161	C₂×C₄	—	28	4	no	—
162	C₂×C₄	—	56	2	no	—
163	C₂×C₄	—	56	2	no	—
164	C₂×C₄	—	56	2	no	—
165	C₂×C₄	—	56	2	no	—
166	C₂×C₄	—	56	2	no	—
167	C₂×C₄	#6432	1	1	yes	characteristic
168	C₂×C₄	C₄×D₂₈	1	1	yes	characteristic
169	C₂×C₄	#6467	1	1	yes	characteristic
170	C₂×C₄	#6467	1	1	yes	characteristic
171	C₂×C₄	C₂×Dic₂₈	1	1	yes	characteristic
172	C₂×C₄	#6493	1	1	yes	characteristic
173	C₈	—	2	2	no	—
174	C₈	C₄×Dic₁₄	1	1	yes	characteristic
175	C₈	#6433	1	1	yes	characteristic
176	C₈	C₂×Dic₇⋊C₄	1	1	yes	characteristic
177	C₈	#6487	1	1	yes	characteristic
178	C₇	#13167	1	1	yes	characteristic, complemented, Sylow
179	C₂²	#89454	1	1	yes	characteristic
180	C₂²	C₄×D₅₆	1	1	yes	characteristic
181	C₂²	#89680	1	2	yes	—
182	C₂²	#89681	1	2	yes	—
183	C₂²	#89887	1	1	yes	characteristic
184	C₄	—	4	1	no	—
185	C₄	—	4	1	no	—
186	C₄	—	56	4	no	—
187	C₄	—	112	2	no	—
188	C₄	#89455	1	1	yes	characteristic
189	C₄	C₄×Dic₂₈	1	1	yes	characteristic
190	C₄	#89700	1	1	yes	characteristic
191	C₄	#89888	1	1	yes	characteristic
192	C₂	#11726823	1	2	yes	—
193	C₂	#11726831	1	2	yes	—
194	C₂	#11727524	1	1	yes	characteristic
195	C₂	#11729197	1	1	yes	characteristic
196	C₂	#11729201	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂×D₅₆
commutator	C₂×C₇×C₈	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₇
Fitting	C₂×C₇×#346	C₂
socle, prime radical	C₂³×C₇	C₂×D₈
cosocle	C₂²×C₇×C₈	C₂³
hypercenter	C₂×#346	D₇
hypocenter	C₇	#13167
LCS	G ▷ C₂×C₇×C₈ ▷ C₄×C₇ ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×#346

Complements

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

N	H ≅ G/N	type
14. #89300	C₄	semidirect
63. C₇×C₁₆	C₄⋊C₄	semidirect
150. C₁₆	Dic₇⋊C₄	semidirect
178. C₇	#13167	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₇×#346	2
C₂×#89300	2
C₂×#89299	2
#11727524	2
#11725612	2
#13167	7