Subgroups — #13244975

Order 1792.

342 conjugacy classes of subgroups.

The subgroup lattice has 236 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	1	yes	maximal, characteristic
3	#11728494	C₂	1	1	yes	maximal, characteristic
4	#11727504	C₂	1	1	yes	maximal, characteristic
5	#11727483	C₂	1	1	yes	maximal, characteristic
6	#11725542	C₂	1	1	yes	maximal, characteristic
7	#11725542	C₂	1	1	yes	maximal, characteristic
8	C₇×#431	C₂²	1	1	yes	characteristic
9	C₇×#431	C₂²	1	1	yes	characteristic
10	C₂×C₄×C₇×C₈	C₂²	1	1	yes	characteristic
11	C₄²×Dic₇	C₂²	1	1	yes	characteristic
12	#89697	C₂²	1	1	yes	characteristic
13	#89697	C₂²	1	1	yes	characteristic
14	C₂×C₄×C₇⋊C₈	C₂²	1	1	yes	characteristic
15	C₇×#335	C₄	1	2	yes	complemented
16	C₇×#334	C₄	1	2	yes	complemented
17	#89345	C₄	1	2	yes	—
18	#89345	C₄	1	2	yes	—
19	#89253	C₄	1	2	yes	complemented
20	#89252	C₄	1	2	yes	complemented
21	#89250	—	2	1	no	—
22	#89249	—	2	1	no	—
23	#89249	—	2	1	no	—
24	C₈×C₇⋊C₈	—	2	1	no	—
25	#8270	—	7	1	no	maximal, Sylow
26	C₂²×C₇×C₈	C₂×C₄	1	1	yes	characteristic
27	C₂²×C₇×C₈	C₂×C₄	1	1	yes	characteristic
28	C₂×C₄²×C₇	C₂³	1	1	yes	characteristic
29	C₂×C₄×Dic₇	—	2	1	no	—
30	C₂×C₄×Dic₇	—	2	1	no	—
31	C₂×C₄×Dic₇	C₂×C₄	1	2	yes	—
32	C₂²×C₇⋊C₈	C₂×C₄	1	2	yes	—
33	C₇×C₄⋊C₈	C₂×C₄	1	2	yes	—
34	C₇×C₄⋊C₈	C₂×C₄	1	2	yes	—
35	C₇×C₂²⋊C₈	—	2	1	no	—
36	C₇×C₂²⋊C₈	—	2	1	no	—
37	C₄×C₇×C₈	—	2	1	no	—
38	C₄×C₇×C₈	—	2	1	no	—
39	C₄×C₇×C₈	C₂×C₄	1	1	yes	characteristic
40	C₄×C₇×C₈	C₂×C₄	1	1	yes	characteristic
41	C₄×C₇×C₈	C₂×C₄	1	1	yes	characteristic
42	C₄×C₇×C₈	C₂×C₄	1	1	yes	characteristic
43	C₂²⋊(C₇⋊C₈)	—	2	1	no	—
44	C₂²⋊(C₇⋊C₈)	—	2	1	no	—
45	C₄⋊(C₇⋊C₈)	C₂×C₄	1	2	yes	—
46	C₄⋊(C₇⋊C₈)	C₂×C₄	1	2	yes	—
47	C₄×C₇⋊C₈	—	2	1	no	—
48	C₄×C₇⋊C₈	—	2	1	no	—
49	C₄×C₇⋊C₈	D₄	1	1	yes	characteristic
50	C₄×C₇⋊C₈	D₄	1	1	yes	characteristic
51	C₄×C₇⋊C₈	Q₈	1	2	yes	—
52	#1744	—	7	1	no	—
53	#1564	—	7	1	no	—
54	#1564	D₇	1	1	yes	characteristic
55	#1452	—	7	1	no	—
56	#1450	—	7	1	no	—
57	#1276	—	7	1	no	—
58	#1276	—	7	1	no	—
59	C₂²×C₄×C₇	C₂²×C₄	1	1	yes	characteristic
60	C₂²×C₄×C₇	#41	1	1	yes	characteristic
61	C₂²×C₄×C₇	#41	1	1	yes	characteristic
62	C₂²×Dic₇	—	2	2	no	—
63	C₂×C₇×C₈	—	2	2	no	—
64	C₂×C₇×C₈	—	2	2	no	—
65	C₂×C₇×C₈	—	2	2	no	—
66	C₂×C₇×C₈	—	2	2	no	—
67	C₂×C₇×C₈	C₄²	1	1	yes	characteristic
68	C₂×C₇×C₈	C₄²	1	1	yes	characteristic
69	C₂×C₇×C₈	C₄²	1	1	yes	characteristic
70	C₂×C₇×C₈	C₄²	1	1	yes	characteristic
71	C₄²×C₇	C₂²×C₄	1	1	yes	characteristic
72	C₄²×C₇	C₂²×C₄	1	1	yes	characteristic
73	C₄²×C₇	C₂×D₄	1	1	yes	characteristic
74	C₄²×C₇	C₂×Q₈	1	1	yes	characteristic
75	C₄×Dic₇	—	2	4	no	—
76	C₄×Dic₇	—	2	2	no	—
77	C₄×Dic₇	—	2	2	no	—
78	C₄×Dic₇	—	4	2	no	—
79	C₂×C₇⋊C₈	—	2	2	no	—
80	C₂×C₇⋊C₈	—	2	2	no	—
81	C₂×C₇⋊C₈	—	2	2	no	—
82	C₂×C₇⋊C₈	—	2	2	no	—
83	C₂×C₇⋊C₈	C₄⋊C₄	1	2	yes	—
84	C₂×C₇⋊C₈	C₄⋊C₄	1	2	yes	—
85	#431	—	7	1	no	—
86	#431	—	7	1	no	—
87	#431	C₂×D₇	1	1	yes	characteristic
88	#431	C₂×D₇	1	1	yes	characteristic
89	C₂×C₄×C₈	—	7	1	no	—
90	C₂×C₄×C₈	C₂×D₇	1	1	yes	characteristic
91	C₄³	—	7	1	no	—
92	#336	—	7	2	no	—
93	#336	—	7	2	no	—
94	#335	—	7	2	no	—
95	#335	Dic₇	1	2	yes	complemented
96	#334	—	7	2	no	—
97	#334	Dic₇	1	2	yes	complemented
98	#322	—	14	1	no	—
99	#322	—	14	1	no	—
100	#322	—	14	1	no	—
101	C₈²	—	14	1	no	—
102	C₂³×C₇	#117	1	1	yes	characteristic
103	C₂×C₄×C₇	—	2	2	no	—
104	C₂×C₄×C₇	—	2	2	no	—
105	C₂×C₄×C₇	C₂×C₄²	1	1	yes	characteristic
106	C₂×C₄×C₇	C₂×C₄⋊C₄	1	1	yes	characteristic
107	C₂×C₄×C₇	C₄×D₄	1	1	yes	characteristic
108	C₂×C₄×C₇	C₄×D₄	1	1	yes	characteristic
109	C₂×C₄×C₇	C₄×Q₈	1	1	yes	characteristic
110	C₂×C₄×C₇	C₄×Q₈	1	1	yes	characteristic
111	C₂×C₄×C₇	#131	1	2	yes	—
112	C₂×C₄×C₇	#131	1	2	yes	—
113	C₂×Dic₇	—	2	4	no	—
114	C₂×Dic₇	—	4	2	no	—
115	C₂×Dic₇	—	4	2	no	—
116	C₇×C₈	—	2	1	no	—
117	C₇×C₈	—	2	1	no	—
118	C₇×C₈	—	2	1	no	—
119	C₇×C₈	—	2	1	no	—
120	C₇×C₈	—	4	2	no	—
121	C₇×C₈	—	4	2	no	—
122	C₇⋊C₈	—	2	2	no	—
123	C₇⋊C₈	—	2	2	no	—
124	C₇⋊C₈	—	4	2	no	—
125	C₇⋊C₈	—	4	2	no	—
126	C₂²×C₈	—	7	2	no	—
127	C₂²×C₈	C₄×D₇	1	1	yes	characteristic
128	C₂²×C₈	C₄×D₇	1	1	yes	characteristic
129	C₂×C₄²	—	7	2	no	—
130	C₂×C₄²	—	14	1	no	—
131	C₂×C₄²	—	14	1	no	—
132	C₂×C₄²	C₂²×D₇	1	1	yes	characteristic
133	C₄⋊C₈	—	7	2	no	—
134	C₄⋊C₈	—	7	2	no	—
135	C₄⋊C₈	C₂×Dic₇	1	2	yes	—
136	C₄⋊C₈	C₂×Dic₇	1	2	yes	—
137	C₂²⋊C₈	—	2	1	no	—
138	C₂²⋊C₈	—	2	1	no	—
139	C₂²⋊C₈	—	14	1	no	—
140	C₂²⋊C₈	—	14	1	no	—
141	C₄×C₈	—	2	1	no	—
142	C₄×C₈	—	2	1	no	—
143	C₄×C₈	—	7	1	no	—
144	C₄×C₈	—	7	1	no	—
145	C₄×C₈	—	7	2	no	—
146	C₄×C₈	—	14	1	no	—
147	C₄×C₈	—	14	1	no	—
148	C₄×C₈	C₄×D₇	1	1	yes	characteristic
149	C₄×C₈	C₄×D₇	1	1	yes	characteristic
150	C₄×C₈	C₂×Dic₇	1	1	yes	characteristic
151	C₄×C₈	C₂×Dic₇	1	1	yes	characteristic
152	C₂²×C₇	—	2	1	no	—
153	C₂²×C₇	—	2	1	no	—
154	C₂²×C₇	C₄×C₄⋊C₄	1	1	yes	characteristic
155	C₂²×C₇	#433	1	2	yes	—
156	C₄×C₇	—	2	1	no	—
157	C₄×C₇	—	2	1	no	—
158	C₄×C₇	#405	1	1	yes	characteristic
159	C₄×C₇	#405	1	1	yes	characteristic
160	C₄×C₇	#424	1	2	yes	—
161	C₄×C₇	#443	1	2	yes	—
162	C₄×C₇	#444	1	2	yes	—
163	Dic₇	—	4	4	no	—
164	C₂²×C₄	—	14	2	no	—
165	C₂²×C₄	C₂×C₄×D₇	1	1	yes	characteristic
166	C₂²×C₄	#1159	1	1	yes	characteristic
167	C₂²×C₄	#1161	1	1	yes	characteristic
168	C₂×C₈	—	2	2	no	—
169	C₂×C₈	—	2	2	no	—
170	C₂×C₈	—	2	2	no	—
171	C₂×C₈	—	2	2	no	—
172	C₂×C₈	—	7	2	no	—
173	C₂×C₈	—	7	2	no	—
174	C₂×C₈	—	14	2	no	—
175	C₂×C₈	—	14	2	no	—
176	C₂×C₈	—	14	2	no	—
177	C₂×C₈	—	14	2	no	—
178	C₂×C₈	C₄×Dic₇	1	1	yes	characteristic
179	C₂×C₈	C₄×Dic₇	1	1	yes	characteristic
180	C₂×C₈	C₄×Dic₇	1	1	yes	characteristic
181	C₂×C₈	C₄×Dic₇	1	1	yes	characteristic
182	C₄²	—	14	4	no	—
183	C₄²	—	14	2	no	—
184	C₄²	—	14	2	no	—
185	C₄²	—	28	2	no	—
186	C₄²	C₂×C₄×D₇	1	1	yes	characteristic
187	C₄²	D₄×D₇	1	1	yes	characteristic
188	C₄²	Q₈×D₇	1	1	yes	characteristic
189	C₄²	C₂²×Dic₇	1	1	yes	characteristic
190	C₂×C₇	—	2	2	no	—
191	C₂×C₇	#1768	1	1	yes	characteristic
192	C₂×C₇	#1772	1	1	yes	characteristic
193	C₂×C₇	#1773	1	1	yes	characteristic
194	C₂³	#6455	1	1	yes	characteristic
195	C₂×C₄	—	2	2	no	—
196	C₂×C₄	—	2	2	no	—
197	C₂×C₄	—	14	4	no	—
198	C₂×C₄	—	28	2	no	—
199	C₂×C₄	—	28	2	no	—
200	C₂×C₄	#6450	1	1	yes	characteristic
201	C₂×C₄	D₇×C₄⋊C₄	1	1	yes	characteristic
202	C₂×C₄	#6456	1	1	yes	characteristic
203	C₂×C₄	#6464	1	2	yes	—
204	C₂×C₄	#6470	1	2	yes	—
205	C₂×C₄	C₂×C₄×Dic₇	1	1	yes	characteristic
206	C₂×C₄	D₄×Dic₇	1	1	yes	characteristic
207	C₂×C₄	Q₈×Dic₇	1	1	yes	characteristic
208	C₈	—	2	1	no	—
209	C₈	—	2	1	no	—
210	C₈	—	2	1	no	—
211	C₈	—	2	1	no	—
212	C₈	—	4	2	no	—
213	C₈	—	4	2	no	—
214	C₈	—	14	2	no	—
215	C₈	—	14	2	no	—
216	C₈	—	28	2	no	—
217	C₈	—	28	2	no	—
218	C₇	#8270	1	1	yes	characteristic, complemented, Sylow
219	C₂²	—	2	1	no	—
220	C₂²	—	2	1	no	—
221	C₂²	#89612	1	2	yes	—
222	C₂²	C₄⋊C₄×Dic₇	1	1	yes	characteristic
223	C₄	—	2	1	no	—
224	C₄	—	2	1	no	—
225	C₄	—	28	4	no	—
226	C₄	#89611	1	2	yes	—
227	C₄	#89667	1	2	yes	—
228	C₄	#89873	1	1	yes	characteristic
229	C₄	#89894	1	1	yes	characteristic
230	C₄	#89969	1	2	yes	—
231	C₂	—	2	2	no	—
232	C₂	#11728196	1	1	yes	characteristic
233	C₂	#11728414	1	1	yes	characteristic
234	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	#8270
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
15. C₇×#335	C₄	semidirect
16. C₇×#334	C₄	semidirect
19. #89253	C₄	semidirect
20. #89252	C₄	semidirect
95. #335	Dic₇	semidirect
97. #334	Dic₇	semidirect
218. C₇	#8270	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
#11728494	2
#11727504	2
#11727483	2
#11725542	2
#8270	7