Subgroups — #13246835

Order 1792.

336 conjugacy classes of subgroups.

The subgroup lattice has 246 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₇×#2960	C₂	1	1	yes	maximal, characteristic
2	#11729588	C₂	1	2	yes	maximal
3	#11727485	C₂	1	1	yes	maximal, characteristic
4	#11727484	C₂	1	1	yes	maximal, characteristic
5	C₇⋊#1309	C₂	1	2	yes	maximal
6	C₇×#533	C₂²	1	1	yes	characteristic
7	C₇×#446	—	2	1	no	—
8	C₇×#446	—	2	1	no	—
9	C₇×C₈×Q₈	—	2	1	no	—
10	C₇×C₈×Q₈	—	2	1	no	—
11	C₇×#424	C₄	1	2	yes	—
12	C₇×#423	C₂²	1	1	yes	characteristic
13	C₇×#405	C₄	1	2	yes	—
14	C₂×C₄×C₇×C₈	C₂²	1	1	yes	characteristic
15	#89696	C₂²	1	2	yes	—
16	#89696	C₂²	1	2	yes	—
17	#89342	—	2	2	no	—
18	#89342	—	2	2	no	—
19	#89253	—	2	1	no	—
20	#89253	—	2	1	no	—
21	#89252	—	2	1	no	—
22	#89252	—	2	1	no	—
23	#8641	—	7	1	no	maximal, Sylow
24	C₂×C₇×M₁₆	C₂×C₄	1	2	yes	—
25	C₂²×C₇×C₈	C₂×C₄	1	1	yes	characteristic
26	C₂²×C₇×C₈	C₂×C₄	1	1	yes	characteristic
27	C₇×C₄⋊Q₈	D₄	1	1	yes	characteristic
28	C₇×C₄⋊Q₈	D₄	1	1	yes	characteristic
29	C₇×#125	Q₈	1	2	yes	—
30	C₇×C₂²⋊Q₈	—	2	1	no	—
31	C₇×C₂²⋊Q₈	—	2	1	no	—
32	C₄×C₇×Q₈	—	2	1	no	—
33	C₄×C₇×Q₈	—	2	1	no	—
34	C₇×#117	C₂×C₄	1	2	yes	—
35	C₂×C₄²×C₇	C₂³	1	1	yes	characteristic
36	C₂×#1136	—	2	2	no	—
37	C₂×#1136	—	2	2	no	—
38	C₇×C₄⋊C₈	—	2	2	no	—
39	C₇×C₄⋊C₈	—	2	2	no	—
40	C₇×C₄⋊C₈	—	2	1	no	—
41	C₇×C₄⋊C₈	—	2	1	no	—
42	C₇×#97	—	2	2	no	—
43	C₄×C₇×C₈	—	2	2	no	—
44	C₄×C₇×C₈	D₄	1	1	yes	characteristic
45	C₄×C₇×C₈	D₄	1	1	yes	characteristic
46	C₄×C₇×C₈	D₄	1	1	yes	characteristic
47	C₄×C₇×C₈	D₄	1	1	yes	characteristic
48	C₄⋊(C₇⋊C₈)	—	2	2	no	—
49	C₄⋊(C₇⋊C₈)	—	2	2	no	—
50	C₄⋊(C₇⋊C₈)	—	2	2	no	—
51	C₄⋊(C₇⋊C₈)	—	2	2	no	—
52	#2960	D₇	1	1	yes	characteristic
53	#1567	—	7	1	no	—
54	#1566	—	7	1	no	—
55	#1561	—	7	2	no	—
56	#1309	—	7	2	no	—
57	C₂×C₇×Q₈	—	2	1	no	—
58	C₂×C₇×Q₈	—	2	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	C₇×M₁₆	—	4	2	no	—
63	C₂×C₇×C₈	—	2	2	no	—
64	C₂×C₇×C₈	—	2	2	no	—
65	C₂×C₇×C₈	—	4	1	no	—
66	C₂×C₇×C₈	C₂²⋊C₄	1	1	yes	characteristic
67	C₂×C₇×C₈	C₂²⋊C₄	1	1	yes	characteristic
68	C₂×C₇×C₈	C₂²⋊C₄	1	1	yes	characteristic
69	C₂×C₇×C₈	C₂²⋊C₄	1	1	yes	characteristic
70	C₇×C₄⋊C₄	—	2	1	no	—
71	C₇×C₄⋊C₄	—	2	1	no	—
72	C₇×C₄⋊C₄	—	2	2	no	—
73	C₇×C₄⋊C₄	—	2	2	no	—
74	C₇×C₄⋊C₄	C₄⋊C₄	1	2	yes	—
75	C₇×C₄⋊C₄	C₄⋊C₄	1	2	yes	—
76	C₇×C₂²⋊C₄	—	2	2	no	—
77	C₄²×C₇	—	2	2	no	—
78	C₄²×C₇	C₂×D₄	1	1	yes	characteristic
79	C₄²×C₇	C₂×D₄	1	1	yes	characteristic
80	C₄²×C₇	C₂×D₄	1	1	yes	characteristic
81	C₄²×C₇	C₂×Q₈	1	1	yes	characteristic
82	#1136	—	8	2	no	—
83	#1136	—	8	2	no	—
84	C₂×C₇⋊C₈	—	4	2	no	—
85	C₂×C₇⋊C₈	—	4	2	no	—
86	#533	C₂×D₇	1	1	yes	characteristic
87	#446	—	2	1	no	—
88	#446	—	2	1	no	—
89	C₈×Q₈	—	2	1	no	—
90	C₈×Q₈	—	2	1	no	—
91	#424	Dic₇	1	2	yes	—
92	#423	—	7	2	no	—
93	#423	—	7	2	no	—
94	#423	C₂×D₇	1	1	yes	characteristic
95	#405	Dic₇	1	2	yes	—
96	C₂×C₄×C₈	C₂×D₇	1	1	yes	characteristic
97	#335	—	14	1	no	—
98	#335	—	14	1	no	—
99	#334	—	14	1	no	—
100	#334	—	14	1	no	—
101	#332	—	14	2	no	—
102	#332	—	14	2	no	—
103	C₂³×C₇	#117	1	1	yes	characteristic
104	C₇×Q₈	—	4	2	no	—
105	C₇×Q₈	—	4	1	no	—
106	C₇×Q₈	—	4	1	no	—
107	C₂×C₄×C₇	—	2	1	no	—
108	C₂×C₄×C₇	—	2	1	no	—
109	C₂×C₄×C₇	—	2	2	no	—
110	C₂×C₄×C₇	—	2	1	no	—
111	C₂×C₄×C₇	—	2	1	no	—
112	C₂×C₄×C₇	—	2	2	no	—
113	C₂×C₄×C₇	—	2	2	no	—
114	C₂×C₄×C₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
115	C₂×C₄×C₇	C₂×C₄⋊C₄	1	1	yes	characteristic
116	C₂×C₄×C₇	#121	1	1	yes	characteristic
117	C₂×C₄×C₇	#121	1	1	yes	characteristic
118	C₂×C₄×C₇	C₂²⋊Q₈	1	1	yes	characteristic
119	C₂×C₄×C₇	C₂²⋊Q₈	1	1	yes	characteristic
120	C₇×C₈	—	2	1	no	—
121	C₇×C₈	—	2	1	no	—
122	C₇×C₈	—	2	1	no	—
123	C₇×C₈	—	4	2	no	—
124	C₇×C₈	#101	1	2	yes	—
125	C₇⋊C₈	—	8	2	no	—
126	C₇⋊C₈	—	8	2	no	—
127	C₂×M₁₆	—	14	2	no	—
128	C₂×M₁₆	—	14	2	no	—
129	C₂×M₁₆	C₂×Dic₇	1	2	yes	—
130	C₂²×C₈	C₂×Dic₇	1	1	yes	characteristic
131	C₂²×C₈	C₂×Dic₇	1	1	yes	characteristic
132	C₄⋊Q₈	D₂₈	1	1	yes	characteristic
133	C₄⋊Q₈	D₂₈	1	1	yes	characteristic
134	#125	Dic₁₄	1	2	yes	—
135	C₂²⋊Q₈	—	2	1	no	—
136	C₂²⋊Q₈	—	2	1	no	—
137	C₄×Q₈	—	2	1	no	—
138	C₄×Q₈	—	2	1	no	—
139	#117	C₂×Dic₇	1	2	yes	—
140	C₂×C₄²	C₂²×D₇	1	1	yes	characteristic
141	C₄⋊C₈	—	2	2	no	—
142	C₄⋊C₈	—	2	2	no	—
143	C₄⋊C₈	—	2	1	no	—
144	C₄⋊C₈	—	2	1	no	—
145	C₄⋊C₈	—	14	2	no	—
146	C₄⋊C₈	—	14	2	no	—
147	C₄⋊C₈	—	14	2	no	—
148	C₄⋊C₈	—	14	2	no	—
149	#97	—	2	2	no	—
150	C₄×C₈	—	2	2	no	—
151	C₄×C₈	C₂²⋊D₇	1	1	yes	characteristic
152	C₄×C₈	C₂²⋊D₇	1	1	yes	characteristic
153	C₄×C₈	C₂²⋊D₇	1	1	yes	characteristic
154	C₄×C₈	C₂²⋊D₇	1	1	yes	characteristic
155	C₂²×C₇	—	2	1	no	—
156	C₂²×C₇	—	2	1	no	—
157	C₂²×C₇	—	2	1	no	—
158	C₂²×C₇	#380	1	1	yes	characteristic
159	C₄×C₇	—	2	1	no	—
160	C₄×C₇	—	2	1	no	—
161	C₄×C₇	—	2	1	no	—
162	C₄×C₇	—	4	2	no	—
163	C₄×C₇	—	4	2	no	—
164	C₄×C₇	C₂×#101	1	1	yes	characteristic
165	C₄×C₇	#413	1	1	yes	characteristic
166	C₄×C₇	#471	1	2	yes	—
167	C₄×C₇	#472	1	1	yes	characteristic
168	C₄×C₇	#473	1	1	yes	characteristic
169	C₂×Q₈	—	2	1	no	—
170	C₂×Q₈	—	2	1	no	—
171	C₂²×C₄	#1157	1	1	yes	characteristic
172	C₂²×C₄	#1157	1	1	yes	characteristic
173	C₂²×C₄	C₂²×Dic₇	1	1	yes	characteristic
174	M₁₆	—	4	2	no	—
175	M₁₆	—	56	2	no	—
176	M₁₆	—	56	2	no	—
177	C₂×C₈	—	2	2	no	—
178	C₂×C₈	—	2	2	no	—
179	C₂×C₈	—	4	1	no	—
180	C₂×C₈	—	28	2	no	—
181	C₂×C₈	—	28	2	no	—
182	C₂×C₈	C₂²⋊Dic₇	1	1	yes	characteristic
183	C₂×C₈	C₂²⋊Dic₇	1	1	yes	characteristic
184	C₂×C₈	C₂²⋊Dic₇	1	1	yes	characteristic
185	C₂×C₈	C₂²⋊Dic₇	1	1	yes	characteristic
186	C₄⋊C₄	—	2	2	no	—
187	C₄⋊C₄	—	2	2	no	—
188	C₄⋊C₄	—	2	1	no	—
189	C₄⋊C₄	—	2	1	no	—
190	C₄⋊C₄	C₄⋊Dic₇	1	2	yes	—
191	C₄⋊C₄	C₄⋊Dic₇	1	2	yes	—
192	C₂²⋊C₄	—	2	2	no	—
193	C₄²	—	2	2	no	—
194	C₄²	C₂×Dic₁₄	1	1	yes	characteristic
195	C₄²	C₂×D₂₈	1	1	yes	characteristic
196	C₄²	C₂×C₂²⋊D₇	1	1	yes	characteristic
197	C₄²	C₂×C₂²⋊D₇	1	1	yes	characteristic
198	C₂×C₇	—	4	1	no	—
199	C₂×C₇	#1801	1	1	yes	characteristic
200	C₂×C₇	#1815	1	1	yes	characteristic
201	C₂×C₇	#1816	1	1	yes	characteristic
202	C₂³	#6489	1	1	yes	characteristic
203	Q₈	—	4	2	no	—
204	Q₈	—	4	1	no	—
205	Q₈	—	4	1	no	—
206	C₂×C₄	—	2	2	no	—
207	C₂×C₄	—	2	1	no	—
208	C₂×C₄	—	2	1	no	—
209	C₂×C₄	—	2	1	no	—
210	C₂×C₄	—	2	1	no	—
211	C₂×C₄	—	2	2	no	—
212	C₂×C₄	—	2	2	no	—
213	C₂×C₄	#6487	1	1	yes	characteristic
214	C₂×C₄	#6487	1	1	yes	characteristic
215	C₂×C₄	C₂×C₄⋊Dic₇	1	1	yes	characteristic
216	C₂×C₄	#6493	1	1	yes	characteristic
217	C₂×C₄	#6493	1	1	yes	characteristic
218	C₂×C₄	C₂×C₂²⋊Dic₇	1	1	yes	characteristic
219	C₈	—	2	1	no	—
220	C₈	—	2	1	no	—
221	C₈	—	2	1	no	—
222	C₈	—	4	2	no	—
223	C₈	—	56	2	no	—
224	C₈	—	56	2	no	—
225	C₈	#6410	1	2	yes	—
226	C₇	#8641	1	1	yes	characteristic, complemented, Sylow
227	C₂²	—	2	1	no	—
228	C₂²	—	2	1	no	—
229	C₂²	—	2	1	no	—
230	C₂²	#89984	1	1	yes	characteristic
231	C₄	—	2	1	no	—
232	C₄	—	2	1	no	—
233	C₄	—	2	1	no	—
234	C₄	—	4	2	no	—
235	C₄	—	4	2	no	—
236	C₄	#89913	1	2	yes	—
237	C₄	#89914	1	1	yes	characteristic
238	C₄	#89915	1	1	yes	characteristic
239	C₄	C₂×#6410	1	1	yes	characteristic
240	C₄	#90009	1	1	yes	characteristic
241	C₂	—	4	1	no	—
242	C₂	#11729391	1	1	yes	characteristic
243	C₂	#11729769	1	1	yes	characteristic
244	C₂	#11729770	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#89984
commutator	C₂²×C₄×C₇	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₇
Fitting	C₇×#2960	C₂
socle	C₂²×C₇	#380
prime radical	C₂³×C₇	#117
cosocle	C₂×C₄²×C₇	C₂³
hypercenter	#2960	D₇
hypocenter	C₇	#8641
LCS	G ▷ C₂²×C₄×C₇ ▷ C₂²×C₇ ▷ C₇
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #2960

Complements

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

N	H ≅ G/N	type
226. C₇	#8641	semidirect

Maximal subgroups

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

subgroup	index
C₇×#2960	2
#11729588	2
#11727485	2
#11727484	2
C₇⋊#1309	2
#8641	7