Subgroups — #13324657

Order 1792.

544 conjugacy classes of subgroups.

The subgroup lattice has 264 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₇×#3170	C₂	1	1	yes	maximal, characteristic
2	#11740776	C₂	1	2	yes	maximal
3	#11729581	C₂	1	2	yes	maximal
4	#11728658	C₂	1	4	yes	maximal, complemented
5	#11728657	C₂	1	4	yes	maximal, complemented
6	#11727486	C₂	1	1	yes	maximal, characteristic
7	#11727480	C₂	1	1	yes	maximal, characteristic
8	C₇×#533	C₂²	1	2	yes	—
9	C₇×#500	C₄	1	1	yes	characteristic
10	C₇×#494	C₄	1	1	yes	characteristic
11	C₇×#487	C₄	1	2	yes	—
12	C₇×#467	C₄	1	2	yes	—
13	C₇×#467	C₄	1	2	yes	—
14	C₄×C₇×Q₁₆	C₂²	1	2	yes	—
15	C₄×C₇×Q₁₆	C₂²	1	2	yes	—
16	C₂×C₄×C₇×C₈	C₂²	1	1	yes	characteristic
17	#89796	—	2	4	no	—
18	#89796	C₂²	1	4	yes	—
19	Q₈×C₇⋊C₈	C₂²	1	4	yes	—
20	#89772	—	2	4	no	—
21	#89698	C₂²	1	2	yes	—
22	#89697	C₂²	1	1	yes	characteristic
23	#89697	C₂²	1	1	yes	characteristic
24	Q₈⋊(C₇⋊C₈)	C₂²	1	8	yes	—
25	#89253	C₂²	1	2	yes	—
26	#89252	C₂²	1	2	yes	—
27	#89249	C₂²	1	2	yes	—
28	#89249	C₂²	1	2	yes	—
29	#17991	—	7	1	no	maximal, Sylow
30	C₂×C₇×Q₁₆	C₂×C₄	1	2	yes	—
31	C₂×C₇×Q₁₆	C₂×C₄	1	2	yes	—
32	C₂²×C₇×C₈	C₂×C₄	1	1	yes	characteristic
33	C₂²×C₇×C₈	C₂×C₄	1	1	yes	characteristic
34	C₇×C₄⋊Q₈	C₂×C₄	1	2	yes	—
35	C₇×C₄⋊Q₈	C₂×C₄	1	2	yes	—
36	C₇×#125	—	2	2	no	—
37	C₇×C₂²⋊Q₈	—	2	2	no	—
38	C₇×C₂²⋊Q₈	C₂×C₄	1	4	yes	—
39	C₄×C₇×Q₈	—	2	2	no	—
40	C₄×C₇×Q₈	C₂³	1	4	yes	—
41	C₇×#117	—	2	2	no	—
42	C₂×C₄²×C₇	C₂³	1	1	yes	characteristic
43	C₇×#107	C₂×C₄	1	2	yes	—
44	C₇×#107	C₂×C₄	1	2	yes	—
45	C₇×#103	C₂×C₄	1	4	yes	—
46	C₇×#103	C₂×C₄	1	4	yes	—
47	C₄×C₇×C₈	C₂×C₄	1	1	yes	characteristic
48	C₄×C₇×C₈	C₂×C₄	1	1	yes	characteristic
49	C₄×C₇×C₈	C₂³	1	1	yes	characteristic
50	C₄×C₇×C₈	C₂³	1	1	yes	characteristic
51	C₂²⋊(C₇⋊C₈)	—	2	2	no	—
52	C₂²⋊(C₇⋊C₈)	—	2	1	no	—
53	C₂²⋊(C₇⋊C₈)	D₄	1	2	yes	—
54	C₄⋊(C₇⋊C₈)	—	2	1	no	—
55	C₄⋊(C₇⋊C₈)	—	2	4	no	—
56	C₄⋊(C₇⋊C₈)	—	2	4	no	—
57	C₄⋊(C₇⋊C₈)	D₄	1	2	yes	—
58	C₄⋊(C₇⋊C₈)	C₂³	1	4	yes	—
59	#6377	—	2	2	no	—
60	#6377	—	2	4	no	—
61	C₄×C₇⋊C₈	—	2	4	no	—
62	C₄×C₇⋊C₈	C₂³	1	2	yes	—
63	C₄×C₇⋊C₈	C₂³	1	2	yes	—
64	#3170	D₇	1	1	yes	characteristic
65	#2985	—	7	2	no	—
66	#1587	—	7	4	no	—
67	#1584	—	7	4	no	—
68	#1568	—	7	1	no	—
69	#1497	—	7	2	no	—
70	#1452	—	7	1	no	—
71	C₂×C₇×Q₈	—	2	2	no	—
72	C₂×C₇×Q₈	C₂²×C₄	1	4	yes	—
73	C₂²×C₄×C₇	C₂²×C₄	1	1	yes	characteristic
74	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
75	C₂²×C₄×C₇	#41	1	1	yes	characteristic
76	C₇×Q₁₆	—	4	2	no	—
77	C₇×Q₁₆	—	4	2	no	—
78	C₂×C₇×C₈	—	4	1	no	—
79	C₂×C₇×C₈	—	4	1	no	—
80	C₂×C₇×C₈	C₂²×C₄	1	1	yes	characteristic
81	C₂×C₇×C₈	C₂²×C₄	1	1	yes	characteristic
82	C₂×C₇×C₈	C₂²×C₄	1	1	yes	characteristic
83	C₂×C₇×C₈	C₂²×C₄	1	1	yes	characteristic
84	C₇×C₄⋊C₄	—	2	2	no	—
85	C₇×C₄⋊C₄	—	2	2	no	—
86	C₇×C₄⋊C₄	—	2	2	no	—
87	C₇×C₄⋊C₄	—	2	4	no	—
88	C₇×C₄⋊C₄	—	2	4	no	—
89	C₇×C₄⋊C₄	C₂²×C₄	1	4	yes	—
90	C₇×C₂²⋊C₄	—	2	4	no	—
91	C₄²×C₇	—	2	4	no	—
92	C₄²×C₇	C₂²×C₄	1	1	yes	characteristic
93	C₄²×C₇	C₂×D₄	1	1	yes	characteristic
94	C₄²×C₇	#41	1	1	yes	characteristic
95	C₄²×C₇	C₂⁴	1	1	yes	characteristic
96	C₂×C₇⋊C₈	—	2	4	no	—
97	C₂×C₇⋊C₈	—	2	2	no	—
98	C₂×C₇⋊C₈	C₂×D₄	1	4	yes	—
99	#533	C₂×D₇	1	2	yes	—
100	#500	Dic₇	1	1	yes	characteristic
101	#494	Dic₇	1	1	yes	characteristic
102	#487	Dic₇	1	2	yes	—
103	#467	Dic₇	1	2	yes	—
104	#467	Dic₇	1	2	yes	—
105	#446	—	7	4	no	—
106	#446	—	14	4	no	—
107	C₈×Q₈	—	7	4	no	—
108	C₄×Q₁₆	C₂×D₇	1	2	yes	—
109	C₄×Q₁₆	C₂×D₇	1	2	yes	—
110	#433	—	14	4	no	—
111	#432	—	7	2	no	—
112	#431	—	7	1	no	—
113	#431	—	7	1	no	—
114	C₂×C₄×C₈	C₂×D₇	1	1	yes	characteristic
115	#335	—	7	2	no	—
116	#334	—	7	2	no	—
117	Q₈⋊C₈	—	7	8	no	—
118	#322	—	7	2	no	—
119	#322	—	7	2	no	—
120	C₂³×C₇	C₄×D₄	1	1	yes	characteristic
121	C₇×Q₈	—	2	4	no	—
122	C₇×Q₈	—	2	4	no	—
123	C₇×Q₈	—	8	2	no	—
124	C₂×C₄×C₇	—	2	1	no	—
125	C₂×C₄×C₇	—	2	1	no	—
126	C₂×C₄×C₇	—	2	2	no	—
127	C₂×C₄×C₇	—	2	4	no	—
128	C₂×C₄×C₇	—	2	4	no	—
129	C₂×C₄×C₇	—	4	1	no	—
130	C₂×C₄×C₇	C₄×D₄	1	1	yes	characteristic
131	C₂×C₄×C₇	C₄×D₄	1	1	yes	characteristic
132	C₂×C₄×C₇	C₄×D₄	1	1	yes	characteristic
133	C₂×C₄×C₇	C₂³×C₄	1	1	yes	characteristic
134	C₂×C₄×C₇	C₂²×D₄	1	1	yes	characteristic
135	C₂×C₄×C₇	C₂×#41	1	1	yes	characteristic
136	C₇×C₈	—	2	1	no	—
137	C₇×C₈	—	2	1	no	—
138	C₇×C₈	—	2	1	no	—
139	C₇×C₈	—	2	1	no	—
140	C₇⋊C₈	—	2	4	no	—
141	C₇⋊C₈	—	4	2	no	—
142	C₇⋊C₈	—	4	4	no	—
143	C₂×Q₁₆	C₂×Dic₇	1	2	yes	—
144	C₂×Q₁₆	C₂×Dic₇	1	2	yes	—
145	C₂²×C₈	C₂×Dic₇	1	1	yes	characteristic
146	C₂²×C₈	C₂×Dic₇	1	1	yes	characteristic
147	C₄⋊Q₈	C₂×Dic₇	1	2	yes	—
148	C₄⋊Q₈	C₂×Dic₇	1	2	yes	—
149	#125	—	2	2	no	—
150	C₂²⋊Q₈	—	2	2	no	—
151	C₂²⋊Q₈	C₂×Dic₇	1	4	yes	—
152	C₄×Q₈	—	2	2	no	—
153	C₄×Q₈	C₂²×D₇	1	4	yes	—
154	#117	—	2	2	no	—
155	C₂×C₄²	C₂²×D₇	1	1	yes	characteristic
156	#107	C₂×Dic₇	1	2	yes	—
157	#107	C₂×Dic₇	1	2	yes	—
158	C₄⋊C₈	—	7	2	no	—
159	C₄⋊C₈	—	7	4	no	—
160	C₄⋊C₈	—	14	4	no	—
161	C₄⋊C₈	—	14	4	no	—
162	C₄⋊C₈	—	14	1	no	—
163	#103	C₂×Dic₇	1	4	yes	—
164	#103	C₂×Dic₇	1	4	yes	—
165	C₂²⋊C₈	—	7	2	no	—
166	C₂²⋊C₈	—	14	2	no	—
167	C₂²⋊C₈	—	14	1	no	—
168	#97	—	14	4	no	—
169	#97	—	14	2	no	—
170	C₄×C₈	—	7	2	no	—
171	C₄×C₈	—	7	2	no	—
172	C₄×C₈	—	14	4	no	—
173	C₄×C₈	C₂×Dic₇	1	1	yes	characteristic
174	C₄×C₈	C₂×Dic₇	1	1	yes	characteristic
175	C₄×C₈	C₂²×D₇	1	1	yes	characteristic
176	C₄×C₈	C₂²×D₇	1	1	yes	characteristic
177	C₂²×C₇	—	2	1	no	—
178	C₂²×C₇	—	2	1	no	—
179	C₂²×C₇	—	2	1	no	—
180	C₂²×C₇	C₂×C₄×D₄	1	1	yes	characteristic
181	C₄×C₇	—	2	1	no	—
182	C₄×C₇	—	2	1	no	—
183	C₄×C₇	—	2	1	no	—
184	C₄×C₇	—	2	1	no	—
185	C₄×C₇	—	4	4	no	—
186	C₄×C₇	—	4	4	no	—
187	C₄×C₇	#568	1	1	yes	characteristic
188	C₄×C₇	#568	1	1	yes	characteristic
189	C₄×C₇	#578	1	2	yes	—
190	C₂×Q₈	—	2	2	no	—
191	C₂×Q₈	C₂²×Dic₇	1	4	yes	—
192	C₂²×C₄	D₄×D₇	1	1	yes	characteristic
193	C₂²×C₄	#1159	1	1	yes	characteristic
194	C₂²×C₄	C₂²×Dic₇	1	1	yes	characteristic
195	Q₁₆	—	4	2	no	—
196	Q₁₆	—	4	2	no	—
197	C₂×C₈	—	4	1	no	—
198	C₂×C₈	—	4	1	no	—
199	C₂×C₈	—	7	4	no	—
200	C₂×C₈	—	14	4	no	—
201	C₂×C₈	—	14	2	no	—
202	C₂×C₈	C₂²×Dic₇	1	1	yes	characteristic
203	C₂×C₈	C₂²×Dic₇	1	1	yes	characteristic
204	C₂×C₈	C₂²×Dic₇	1	1	yes	characteristic
205	C₂×C₈	C₂²×Dic₇	1	1	yes	characteristic
206	C₄⋊C₄	—	2	2	no	—
207	C₄⋊C₄	—	2	2	no	—
208	C₄⋊C₄	—	2	2	no	—
209	C₄⋊C₄	—	2	4	no	—
210	C₄⋊C₄	—	2	4	no	—
211	C₄⋊C₄	C₂²×Dic₇	1	4	yes	—
212	C₂²⋊C₄	—	2	4	no	—
213	C₄²	—	2	4	no	—
214	C₄²	D₄×D₇	1	1	yes	characteristic
215	C₄²	#1159	1	1	yes	characteristic
216	C₄²	C₂²×Dic₇	1	1	yes	characteristic
217	C₄²	C₂³×D₇	1	1	yes	characteristic
218	C₂×C₇	—	4	1	no	—
219	C₂×C₇	#2932	1	1	yes	characteristic
220	C₂×C₇	#2950	1	1	yes	characteristic
221	C₂×C₇	#2957	1	1	yes	characteristic
222	C₂³	D₄×Dic₇	1	1	yes	characteristic
223	Q₈	—	2	4	no	—
224	Q₈	—	2	4	no	—
225	Q₈	—	8	2	no	—
226	C₂×C₄	—	2	2	no	—
227	C₂×C₄	—	2	1	no	—
228	C₂×C₄	—	2	1	no	—
229	C₂×C₄	—	2	4	no	—
230	C₂×C₄	—	2	4	no	—
231	C₂×C₄	—	4	1	no	—
232	C₂×C₄	D₄×Dic₇	1	1	yes	characteristic
233	C₂×C₄	D₄×Dic₇	1	1	yes	characteristic
234	C₂×C₄	D₄×Dic₇	1	1	yes	characteristic
235	C₂×C₄	C₂×D₄×D₇	1	1	yes	characteristic
236	C₂×C₄	C₂×#1159	1	1	yes	characteristic
237	C₂×C₄	C₂³×Dic₇	1	1	yes	characteristic
238	C₈	—	2	1	no	—
239	C₈	—	2	1	no	—
240	C₈	—	2	1	no	—
241	C₈	—	2	1	no	—
242	C₈	—	14	4	no	—
243	C₈	—	28	2	no	—
244	C₈	—	28	4	no	—
245	C₇	#17991	1	1	yes	characteristic, complemented, Sylow
246	C₂²	—	2	1	no	—
247	C₂²	—	2	1	no	—
248	C₂²	—	2	1	no	—
249	C₂²	C₂×D₄×Dic₇	1	1	yes	characteristic
250	C₄	—	2	1	no	—
251	C₄	—	2	1	no	—
252	C₄	—	2	1	no	—
253	C₄	—	2	1	no	—
254	C₄	—	4	4	no	—
255	C₄	—	4	4	no	—
256	C₄	#90457	1	2	yes	—
257	C₄	#90510	1	1	yes	characteristic
258	C₄	#90510	1	1	yes	characteristic
259	C₂	—	4	1	no	—
260	C₂	#11740721	1	1	yes	characteristic
261	C₂	#11741319	1	1	yes	characteristic
262	C₂	#11741355	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×D₄×D₇
commutator	C₂×C₄×C₇	C₂³×C₄
Frattini	C₄²	C₂³×D₇
Fitting	C₇×#3170	C₂
socle	C₂²×C₇	C₂×C₄×D₄
prime radical	C₂³×C₇	C₄×D₄
cosocle	C₄²×C₇	C₂⁴
hypercenter	#3170	D₇
hypocenter	C₇	#17991
LCS	G ▷ C₂×C₄×C₇ ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #3170

Complements

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

N	H ≅ G/N	type
4. #11728658	C₂	semidirect
5. #11728657	C₂	semidirect
245. C₇	#17991	semidirect

Maximal subgroups

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

subgroup	index
C₇×#3170	2
#11740776	2
#11729581	2
#11728658	2
#11728657	2
#11727486	2
#11727480	2
#17991	7