Subgroups — #13506361

Order 1792.

1006 conjugacy classes of subgroups.

The subgroup lattice has 208 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#89627	C₂	1	4	yes	maximal
2	#11731801	C₂	1	2	yes	maximal
3	C₇×#1844	C₂	1	1	yes	maximal, characteristic
4	#11728308	C₂	1	4	yes	maximal
5	#11727518	C₂	1	1	yes	maximal, characteristic
6	C₄×C₈×Dic₇	C₂	1	1	yes	maximal, characteristic
7	#11727330	C₂	1	2	yes	maximal
8	C₂×#6451	—	2	4	no	—
9	C₂×#6451	C₂²	1	4	yes	—
10	C₂×C₇×#106	C₂²	1	4	yes	—
11	C₂×C₄×C₇×C₈	C₂²	1	1	yes	characteristic
12	C₇×#384	C₂²	1	2	yes	—
13	C₂×#6391	C₂²	1	4	yes	—
14	C₂×C₈×Dic₇	C₂²	1	2	yes	—
15	C₂×C₈×Dic₇	C₂²	1	2	yes	—
16	#89748	—	2	4	no	—
17	#89748	C₂²	1	4	yes	—
18	C₂×#6382	C₂²	1	8	yes	—
19	#89707	C₂²	1	2	yes	—
20	C₄²×Dic₇	C₂²	1	1	yes	characteristic
21	C₂×C₄×C₇⋊C₈	C₂²	1	1	yes	characteristic
22	#89627	—	2	16	no	—
23	#33310	—	7	1	no	maximal, Sylow
24	C₂²×Dic₁₄	—	2	4	no	—
25	C₂²×C₇×C₈	C₂³	1	2	yes	—
26	C₂×C₇×C₄⋊C₄	—	2	4	no	—
27	C₂×C₇×C₄⋊C₄	C₂³	1	4	yes	—
28	C₂×C₄²×C₇	C₂³	1	1	yes	characteristic
29	C₂×C₄⋊Dic₇	—	2	4	no	—
30	C₂×C₄⋊Dic₇	C₂³	1	4	yes	—
31	C₂×Dic₇⋊C₄	—	2	8	no	—
32	C₂×Dic₇⋊C₄	—	2	4	no	—
33	C₂×C₄×Dic₇	D₄	1	4	yes	—
34	C₂×C₄×Dic₇	C₂³	1	1	yes	characteristic
35	C₂×C₄×Dic₇	C₂³	1	1	yes	characteristic
36	C₂²×C₇⋊C₈	C₂³	1	2	yes	—
37	#6451	—	2	16	no	—
38	#6451	—	4	16	no	—
39	C₇×#106	—	2	8	no	—
40	C₄×C₇×C₈	Q₈	1	4	yes	complemented
41	#6391	—	2	8	no	—
42	C₈×Dic₇	D₄	1	8	yes	—
43	C₈×Dic₇	Q₈	1	8	yes	complemented
44	#6382	—	2	16	no	—
45	C₄×C₇⋊C₈	Q₈	1	4	yes	complemented
46	C₂×#498	—	7	4	no	—
47	#2868	—	7	2	no	—
48	#1844	—	7	4	no	—
49	#1844	—	7	1	no	—
50	#1844	—	7	2	no	—
51	#1844	D₇	1	1	yes	characteristic
52	C₄²×C₈	—	7	1	no	—
53	C₂²×C₄×C₇	—	2	4	no	—
54	C₂²×C₄×C₇	C₂×D₄	1	2	yes	—
55	C₂²×C₄×C₇	C₂⁴	1	1	yes	characteristic
56	C₂²×Dic₇	—	2	4	no	—
57	C₂²×Dic₇	C₂×D₄	1	2	yes	—
58	C₂²×Dic₇	C₂×D₄	1	2	yes	—
59	C₂×Dic₁₄	—	4	8	no	—
60	C₂×Dic₁₄	—	4	16	no	—
61	C₂×C₇×C₈	C₂×D₄	1	4	yes	—
62	C₂×C₇×C₈	C₂×Q₈	1	4	yes	—
63	C₂×C₇×C₈	C₂×Q₈	1	4	yes	—
64	C₇×C₄⋊C₄	—	2	8	no	—
65	C₇×C₄⋊C₄	—	4	8	no	—
66	C₄²×C₇	C₂×Q₈	1	2	yes	—
67	C₄²×C₇	C₂×Q₈	1	2	yes	—
68	C₄⋊Dic₇	—	2	8	no	—
69	C₄⋊Dic₇	—	4	8	no	—
70	Dic₇⋊C₄	—	4	8	no	—
71	Dic₇⋊C₄	—	4	16	no	—
72	C₄×Dic₇	SD₁₆	1	16	yes	—
73	C₄×Dic₇	C₂×D₄	1	4	yes	—
74	C₄×Dic₇	C₂×Q₈	1	4	yes	—
75	C₂×C₇⋊C₈	C₂×D₄	1	4	yes	—
76	C₂×C₇⋊C₈	C₂×Q₈	1	4	yes	—
77	C₂×C₇⋊C₈	C₂×Q₈	1	4	yes	—
78	C₂×C₄⋊Q₈	—	7	4	no	—
79	C₂×C₄⋊Q₈	—	14	4	no	—
80	#498	—	14	16	no	—
81	C₂×#106	—	7	4	no	—
82	C₂×#106	—	7	8	no	—
83	C₂×#106	C₂×D₇	1	4	yes	—
84	C₂×C₄×C₈	—	7	2	no	—
85	C₂×C₄×C₈	—	7	2	no	—
86	C₂×C₄×C₈	—	7	1	no	—
87	C₂×C₄×C₈	C₂×D₇	1	1	yes	characteristic
88	#384	—	7	4	no	—
89	#384	—	7	2	no	—
90	#384	—	14	4	no	—
91	#384	C₂×D₇	1	2	yes	—
92	C₄³	—	7	1	no	—
93	C₂³×C₇	C₂²×D₄	1	1	yes	characteristic
94	C₂×C₄×C₇	—	4	4	no	—
95	C₂×C₄×C₇	—	4	8	no	—
96	C₂×C₄×C₇	C₄⋊Q₈	1	4	yes	—
97	C₂×C₄×C₇	C₂×SD₁₆	1	8	yes	—
98	C₂×C₄×C₇	C₂²×D₄	1	2	yes	—
99	C₂×C₄×C₇	C₂²×Q₈	1	1	yes	characteristic
100	C₂×C₄×C₇	C₂²×Q₈	1	1	yes	characteristic
101	C₂×C₄×C₇	C₂²×Q₈	1	2	yes	—
102	C₂×Dic₇	—	4	4	no	—
103	C₂×Dic₇	—	4	8	no	—
104	C₂×Dic₇	#127	1	4	yes	—
105	C₂×Dic₇	C₄⋊Q₈	1	4	yes	—
106	C₂×Dic₇	C₂×SD₁₆	1	8	yes	—
107	C₂×Dic₇	C₂×SD₁₆	1	8	yes	—
108	Dic₁₄	—	8	16	no	—
109	C₇×C₈	C₄⋊Q₈	1	8	yes	complemented
110	C₇⋊C₈	C₄⋊Q₈	1	8	yes	complemented
111	C₂²×Q₈	—	14	4	no	—
112	C₂²×C₈	—	7	2	no	—
113	C₂²×C₈	C₂²×D₇	1	2	yes	—
114	C₄⋊Q₈	—	14	16	no	—
115	C₄⋊Q₈	—	28	16	no	—
116	C₂×C₄⋊C₄	—	2	4	no	—
117	C₂×C₄⋊C₄	—	7	4	no	—
118	C₂×C₄⋊C₄	—	14	4	no	—
119	C₂×C₄⋊C₄	—	14	8	no	—
120	C₂×C₄⋊C₄	—	14	4	no	—
121	C₂×C₄⋊C₄	C₂²×D₇	1	4	yes	—
122	C₂×C₄²	—	7	4	no	—
123	C₂×C₄²	—	7	1	no	—
124	C₂×C₄²	—	7	1	no	—
125	C₂×C₄²	C₂²×D₇	1	1	yes	characteristic
126	#106	—	2	8	no	—
127	#106	—	14	16	no	—
128	#106	—	14	8	no	—
129	C₄×C₈	—	7	8	no	—
130	C₄×C₈	—	7	8	no	—
131	C₄×C₈	—	7	4	no	—
132	C₄×C₈	Dic₁₄	1	4	yes	complemented
133	C₂²×C₇	C₂×#127	1	1	yes	characteristic
134	C₂²×C₇	C₂×C₄⋊Q₈	1	1	yes	characteristic
135	C₂²×C₇	C₂×C₄⋊Q₈	1	1	yes	characteristic
136	C₂²×C₇	C₂²×SD₁₆	1	4	yes	—
137	C₄×C₇	—	8	4	no	—
138	C₄×C₇	#498	1	8	yes	complemented
139	C₄×C₇	C₂×C₄⋊Q₈	1	2	yes	—
140	C₄×C₇	C₂×C₄⋊Q₈	1	2	yes	—
141	Dic₇	—	8	4	no	—
142	Dic₇	#492	1	8	yes	—
143	Dic₇	#498	1	8	yes	complemented
144	C₂×Q₈	—	28	8	no	—
145	C₂×Q₈	—	28	16	no	—
146	C₂²×C₄	—	2	4	no	—
147	C₂²×C₄	—	7	2	no	—
148	C₂²×C₄	—	7	2	no	—
149	C₂²×C₄	—	14	4	no	—
150	C₂²×C₄	D₄×D₇	1	2	yes	—
151	C₂²×C₄	C₂³×D₇	1	1	yes	characteristic
152	C₂×C₈	—	7	4	no	—
153	C₂×C₈	—	7	4	no	—
154	C₂×C₈	—	7	4	no	—
155	C₂×C₈	C₂×Dic₁₄	1	4	yes	—
156	C₂×C₈	D₄×D₇	1	4	yes	—
157	C₂×C₈	Q₈×D₇	1	4	yes	—
158	C₄⋊C₄	—	2	8	no	—
159	C₄⋊C₄	—	4	8	no	—
160	C₄⋊C₄	—	14	8	no	—
161	C₄⋊C₄	—	28	8	no	—
162	C₄⋊C₄	—	28	16	no	—
163	C₄⋊C₄	—	28	8	no	—
164	C₄²	—	7	16	no	—
165	C₄²	—	7	4	no	—
166	C₄²	—	7	4	no	—
167	C₄²	C₂×Dic₁₄	1	2	yes	—
168	C₄²	Q₈×D₇	1	2	yes	—
169	C₂×C₇	#2868	1	1	yes	characteristic
170	C₂×C₇	C₂×#492	1	2	yes	—
171	C₂×C₇	C₂×#498	1	2	yes	—
172	C₂×C₇	C₂×#498	1	2	yes	—
173	C₂³	C₂×D₄×D₇	1	1	yes	characteristic
174	Q₈	—	56	16	no	—
175	C₂×C₄	—	4	4	no	—
176	C₂×C₄	—	4	8	no	—
177	C₂×C₄	—	7	4	no	—
178	C₂×C₄	—	7	4	no	—
179	C₂×C₄	—	7	8	no	—
180	C₂×C₄	—	7	8	no	—
181	C₂×C₄	—	28	4	no	—
182	C₂×C₄	—	28	8	no	—
183	C₂×C₄	#6451	1	4	yes	—
184	C₂×C₄	D₇×SD₁₆	1	8	yes	—
185	C₂×C₄	C₂²×Dic₁₄	1	1	yes	characteristic
186	C₂×C₄	C₂×D₄×D₇	1	2	yes	—
187	C₂×C₄	C₂×Q₈×D₇	1	1	yes	characteristic
188	C₂×C₄	C₂×Q₈×D₇	1	2	yes	—
189	C₈	—	7	8	no	—
190	C₈	#6451	1	8	yes	complemented
191	C₇	#33310	1	1	yes	characteristic, complemented, Sylow
192	C₂²	C₂×#6451	1	1	yes	characteristic
193	C₂²	D₇×#127	1	1	yes	characteristic
194	C₂²	D₇×C₄⋊Q₈	1	1	yes	characteristic
195	C₂²	C₂×D₇×SD₁₆	1	4	yes	—
196	C₄	—	7	8	no	—
197	C₄	—	7	8	no	—
198	C₄	—	8	4	no	—
199	C₄	—	56	4	no	—
200	C₄	#89627	1	8	yes	complemented
201	C₄	C₂×#6451	1	2	yes	—
202	C₄	D₇×C₄⋊Q₈	1	2	yes	—
203	C₂	#11731801	1	1	yes	characteristic
204	C₂	C₂×#89627	1	2	yes	—
205	C₂	D₇×#492	1	2	yes	—
206	C₂	D₇×#498	1	2	yes	—

	subgroup	G/H
center	C₂³	C₂×D₄×D₇
commutator, cosocle	C₂²×C₄×C₇	C₂⁴
Frattini	C₂²×C₄	C₂³×D₇
Fitting	C₇×#1844	C₂
socle, prime radical	C₂³×C₇	C₂²×D₄
hypercenter	#1844	D₇
hypocenter	C₇	#33310
LCS	G ▷ C₂²×C₄×C₇ ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #1844

Complements

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

N	H ≅ G/N	type
40. C₄×C₇×C₈	Q₈	semidirect
43. C₈×Dic₇	Q₈	semidirect
45. C₄×C₇⋊C₈	Q₈	semidirect
109. C₇×C₈	C₄⋊Q₈	semidirect
110. C₇⋊C₈	C₄⋊Q₈	semidirect
132. C₄×C₈	Dic₁₄	semidirect
138. C₄×C₇	#498	semidirect
143. Dic₇	#498	semidirect
190. C₈	#6451	semidirect
191. C₇	#33310	semidirect
200. C₄	#89627	semidirect

Maximal subgroups

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

subgroup	index
C₂×#89627	2
#11731801	2
C₇×#1844	2
#11728308	2
#11727518	2
C₄×C₈×Dic₇	2
#11727330	2
#33310	7