Subgroups — #13244960

Order 1792.

390 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	C₇×#1564	C₂	1	1	yes	maximal, characteristic
2	#11728496	C₂	1	2	yes	maximal
3	C₄×C₈×Dic₇	C₂	1	1	yes	maximal, characteristic
4	#11727483	C₂	1	1	yes	maximal, characteristic
5	#11725542	C₂	1	2	yes	maximal
6	C₇×#431	C₂²	1	2	yes	—
7	C₂×C₄×C₇×C₈	C₂²	1	1	yes	characteristic
8	C₂×C₈×Dic₇	C₄	1	2	yes	—
9	C₂×C₈×Dic₇	C₄	1	2	yes	—
10	C₄²×Dic₇	C₂²	1	1	yes	characteristic
11	#89697	C₂²	1	2	yes	—
12	C₂×C₄×C₇⋊C₈	C₂²	1	1	yes	characteristic
13	C₇×#335	C₄	1	2	yes	complemented
14	C₇×#334	C₄	1	2	yes	complemented
15	#89253	C₄	1	2	yes	complemented
16	#89252	C₄	1	2	yes	complemented
17	#89250	—	2	2	no	—
18	#89249	—	2	2	no	—
19	#8267	—	7	1	no	maximal, Sylow
20	C₂²×C₇×C₈	C₂×C₄	1	1	yes	characteristic
21	C₂²×C₇×C₈	C₂×C₄	1	1	yes	characteristic
22	C₂×C₄²×C₇	C₂³	1	1	yes	characteristic
23	C₂×C₄×Dic₇	—	2	1	no	—
24	C₂×C₄×Dic₇	—	2	1	no	—
25	C₂×C₄×Dic₇	C₂×C₄	1	2	yes	—
26	C₂²×C₇⋊C₈	C₂×C₄	1	2	yes	—
27	C₇×C₄⋊C₈	C₂×C₄	1	4	yes	—
28	C₇×C₂²⋊C₈	—	2	2	no	—
29	C₄×C₇×C₈	—	2	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	C₂×C₄	1	1	yes	characteristic
33	C₄×C₇×C₈	C₂×C₄	1	1	yes	characteristic
34	C₂²⋊(C₇⋊C₈)	—	2	2	no	—
35	C₈×Dic₇	—	2	4	no	—
36	C₈×Dic₇	—	2	4	no	—
37	C₄⋊(C₇⋊C₈)	C₂×C₄	1	4	yes	—
38	C₄×C₇⋊C₈	—	2	2	no	—
39	C₄×C₇⋊C₈	D₄	1	1	yes	characteristic
40	C₄×C₇⋊C₈	D₄	1	1	yes	characteristic
41	C₄×C₇⋊C₈	Q₈	1	2	yes	—
42	C₄²×C₈	—	7	1	no	—
43	#1564	—	7	1	no	—
44	#1564	D₇	1	1	yes	characteristic
45	#1452	—	7	2	no	—
46	#1276	—	7	2	no	—
47	C₂²×C₄×C₇	C₂²×C₄	1	1	yes	characteristic
48	C₂²×C₄×C₇	#41	1	1	yes	characteristic
49	C₂²×C₄×C₇	#41	1	1	yes	characteristic
50	C₂²×Dic₇	—	2	2	no	—
51	C₂×C₇×C₈	—	2	2	no	—
52	C₂×C₇×C₈	—	2	4	no	—
53	C₂×C₇×C₈	C₄²	1	1	yes	characteristic
54	C₂×C₇×C₈	C₄²	1	1	yes	characteristic
55	C₂×C₇×C₈	C₄²	1	1	yes	characteristic
56	C₂×C₇×C₈	C₄²	1	1	yes	characteristic
57	C₂×C₇×C₈	M₁₆	1	4	yes	—
58	C₄²×C₇	C₂²×C₄	1	1	yes	characteristic
59	C₄²×C₇	C₂²×C₄	1	1	yes	characteristic
60	C₄²×C₇	C₂×D₄	1	1	yes	characteristic
61	C₄²×C₇	C₂×Q₈	1	1	yes	characteristic
62	C₄×Dic₇	—	2	4	no	—
63	C₄×Dic₇	—	2	4	no	—
64	C₄×Dic₇	—	2	2	no	—
65	C₄×Dic₇	M₁₆	1	4	yes	—
66	C₂×C₇⋊C₈	—	2	2	no	—
67	C₂×C₇⋊C₈	—	2	2	no	—
68	C₂×C₇⋊C₈	—	2	4	no	—
69	C₂×C₇⋊C₈	C₄⋊C₄	1	2	yes	—
70	C₂×C₇⋊C₈	C₄⋊C₄	1	2	yes	—
71	#431	—	7	2	no	—
72	#431	C₂×D₇	1	2	yes	—
73	C₂×C₄×C₈	—	7	1	no	—
74	C₂×C₄×C₈	—	7	2	no	—
75	C₂×C₄×C₈	—	7	2	no	—
76	C₂×C₄×C₈	C₂×D₇	1	1	yes	characteristic
77	C₄³	—	7	1	no	—
78	#335	—	7	2	no	—
79	#335	Dic₇	1	2	yes	complemented
80	#334	—	7	2	no	—
81	#334	Dic₇	1	2	yes	complemented
82	#322	—	14	2	no	—
83	#322	—	14	2	no	—
84	C₂³×C₇	#117	1	1	yes	characteristic
85	C₂×C₄×C₇	—	2	2	no	—
86	C₂×C₄×C₇	—	2	2	no	—
87	C₂×C₄×C₇	C₂×C₄²	1	1	yes	characteristic
88	C₂×C₄×C₇	C₂×C₄⋊C₄	1	1	yes	characteristic
89	C₂×C₄×C₇	C₄×D₄	1	1	yes	characteristic
90	C₂×C₄×C₇	C₄×D₄	1	1	yes	characteristic
91	C₂×C₄×C₇	C₄×Q₈	1	1	yes	characteristic
92	C₂×C₄×C₇	C₄×Q₈	1	1	yes	characteristic
93	C₂×C₄×C₇	C₂×M₁₆	1	2	yes	—
94	C₂×C₄×C₇	C₂×M₁₆	1	2	yes	—
95	C₂×Dic₇	—	2	4	no	—
96	C₂×Dic₇	—	2	4	no	—
97	C₂×Dic₇	—	2	4	no	—
98	C₇×C₈	—	2	1	no	—
99	C₇×C₈	—	2	1	no	—
100	C₇×C₈	—	4	4	no	—
101	C₇×C₈	#97	1	2	yes	—
102	C₇×C₈	#97	1	2	yes	—
103	C₇⋊C₈	—	2	2	no	—
104	C₇⋊C₈	—	2	2	no	—
105	C₇⋊C₈	—	4	4	no	—
106	C₂²×C₈	—	7	2	no	—
107	C₂²×C₈	C₄×D₇	1	1	yes	characteristic
108	C₂²×C₈	C₄×D₇	1	1	yes	characteristic
109	C₂×C₄²	—	7	2	no	—
110	C₂×C₄²	—	14	1	no	—
111	C₂×C₄²	—	14	1	no	—
112	C₂×C₄²	C₂²×D₇	1	1	yes	characteristic
113	C₄⋊C₈	—	7	4	no	—
114	C₄⋊C₈	C₂×Dic₇	1	4	yes	—
115	C₂²⋊C₈	—	2	2	no	—
116	C₂²⋊C₈	—	14	2	no	—
117	C₄×C₈	—	2	2	no	—
118	C₄×C₈	—	7	1	no	—
119	C₄×C₈	—	7	1	no	—
120	C₄×C₈	—	7	2	no	—
121	C₄×C₈	—	14	4	no	—
122	C₄×C₈	—	14	4	no	—
123	C₄×C₈	—	14	2	no	—
124	C₄×C₈	C₄×D₇	1	1	yes	characteristic
125	C₄×C₈	C₄×D₇	1	1	yes	characteristic
126	C₄×C₈	C₂×Dic₇	1	1	yes	characteristic
127	C₄×C₈	C₂×Dic₇	1	1	yes	characteristic
128	C₂²×C₇	—	2	1	no	—
129	C₂²×C₇	—	2	1	no	—
130	C₂²×C₇	C₄×C₄⋊C₄	1	1	yes	characteristic
131	C₂²×C₇	#432	1	2	yes	—
132	C₄×C₇	—	2	1	no	—
133	C₄×C₇	—	2	1	no	—
134	C₄×C₇	C₂×#97	1	1	yes	characteristic
135	C₄×C₇	C₄×M₁₆	1	1	yes	characteristic
136	C₄×C₇	#423	1	2	yes	—
137	C₄×C₇	#443	1	2	yes	—
138	C₄×C₇	#443	1	2	yes	—
139	Dic₇	—	2	8	no	—
140	C₂²×C₄	—	14	2	no	—
141	C₂²×C₄	C₂×C₄×D₇	1	1	yes	characteristic
142	C₂²×C₄	#1159	1	1	yes	characteristic
143	C₂²×C₄	#1161	1	1	yes	characteristic
144	C₂×C₈	—	2	2	no	—
145	C₂×C₈	—	2	4	no	—
146	C₂×C₈	—	7	2	no	—
147	C₂×C₈	—	7	2	no	—
148	C₂×C₈	—	14	2	no	—
149	C₂×C₈	—	14	2	no	—
150	C₂×C₈	—	14	4	no	—
151	C₂×C₈	#1131	1	4	yes	—
152	C₂×C₈	C₄×Dic₇	1	1	yes	characteristic
153	C₂×C₈	C₄×Dic₇	1	1	yes	characteristic
154	C₂×C₈	C₄×Dic₇	1	1	yes	characteristic
155	C₂×C₈	C₄×Dic₇	1	1	yes	characteristic
156	C₄²	—	7	4	no	—
157	C₄²	—	14	4	no	—
158	C₄²	—	14	4	no	—
159	C₄²	—	14	2	no	—
160	C₄²	C₂×C₄×D₇	1	1	yes	characteristic
161	C₄²	D₄×D₇	1	1	yes	characteristic
162	C₄²	Q₈×D₇	1	1	yes	characteristic
163	C₄²	C₂²×Dic₇	1	1	yes	characteristic
164	C₂×C₇	—	2	2	no	—
165	C₂×C₇	#1767	1	1	yes	characteristic
166	C₂×C₇	#1772	1	1	yes	characteristic
167	C₂×C₇	#1772	1	1	yes	characteristic
168	C₂³	#6455	1	1	yes	characteristic
169	C₂×C₄	—	2	2	no	—
170	C₂×C₄	—	2	2	no	—
171	C₂×C₄	—	14	4	no	—
172	C₂×C₄	—	14	4	no	—
173	C₂×C₄	—	14	4	no	—
174	C₂×C₄	#6450	1	1	yes	characteristic
175	C₂×C₄	D₇×C₄⋊C₄	1	1	yes	characteristic
176	C₂×C₄	#6456	1	1	yes	characteristic
177	C₂×C₄	C₂×#1131	1	2	yes	—
178	C₂×C₄	D₇×M₁₆	1	2	yes	—
179	C₂×C₄	C₂×C₄×Dic₇	1	1	yes	characteristic
180	C₂×C₄	D₄×Dic₇	1	1	yes	characteristic
181	C₂×C₄	Q₈×Dic₇	1	1	yes	characteristic
182	C₈	—	2	1	no	—
183	C₈	—	2	1	no	—
184	C₈	—	4	4	no	—
185	C₈	—	14	2	no	—
186	C₈	—	14	2	no	—
187	C₈	—	28	4	no	—
188	C₈	#6389	1	2	yes	—
189	C₈	#6389	1	2	yes	—
190	C₇	#8267	1	1	yes	characteristic, complemented, Sylow
191	C₂²	—	2	1	no	—
192	C₂²	—	2	1	no	—
193	C₂²	#89607	1	2	yes	—
194	C₂²	C₄⋊C₄×Dic₇	1	1	yes	characteristic
195	C₄	—	2	1	no	—
196	C₄	—	2	1	no	—
197	C₄	—	14	8	no	—
198	C₄	#89609	1	2	yes	—
199	C₄	#89667	1	2	yes	—
200	C₄	C₂×#6389	1	1	yes	characteristic
201	C₄	M₁₆×Dic₇	1	1	yes	characteristic
202	C₄	#89968	1	2	yes	—
203	C₂	—	2	2	no	—
204	C₂	#11728195	1	1	yes	characteristic
205	C₂	#11728413	1	1	yes	characteristic
206	C₂	#11728456	1	1	yes	characteristic

	subgroup	G/H
center	C₄²	D₄×D₇
commutator	C₂×C₄×C₇	C₂×C₄²
Frattini	C₂×C₄²	C₂²×D₇
Fitting	C₇×#1564	C₂
socle	C₂²×C₇	C₄×C₄⋊C₄
prime radical	C₂³×C₇	#117
cosocle	C₂×C₄²×C₇	C₂³
hypercenter	#1564	D₇
hypocenter	C₇	#8267
LCS	G ▷ C₂×C₄×C₇ ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₄² ◁ C₂×C₄² ◁ #1564

Complements

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

N	H ≅ G/N	type
13. C₇×#335	C₄	semidirect
14. C₇×#334	C₄	semidirect
15. #89253	C₄	semidirect
16. #89252	C₄	semidirect
79. #335	Dic₇	semidirect
81. #334	Dic₇	semidirect
190. C₇	#8267	semidirect

Maximal subgroups

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

subgroup	index
C₇×#1564	2
#11728496	2
C₄×C₈×Dic₇	2
#11727483	2
#11725542	2
#8267	7