Subgroups — #13324328

Order 1792.

614 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	#11740773	C₂	1	2	yes	maximal
3	#11729580	C₂	1	2	yes	maximal
4	#11728659	C₂	1	4	yes	maximal, complemented
5	Q₁₆×C₇⋊C₈	C₂	1	4	yes	maximal, complemented
6	#11727485	C₂	1	1	yes	maximal, characteristic
7	C₂×C₈×C₇⋊C₈	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	4	yes	—
13	C₄×C₇×Q₁₆	C₂²	1	4	yes	—
14	C₂×C₄×C₇×C₈	C₂²	1	1	yes	characteristic
15	#89796	—	2	2	no	—
16	#89796	C₂²	1	4	yes	—
17	Q₈×C₇⋊C₈	—	2	2	no	—
18	Q₈×C₇⋊C₈	C₂²	1	4	yes	—
19	#89771	—	2	2	no	—
20	#89769	—	2	2	no	—
21	#89696	C₂²	1	2	yes	—
22	C₂×C₄×C₇⋊C₈	C₂²	1	1	yes	characteristic
23	C₂×C₄×C₇⋊C₈	C₂²	1	1	yes	characteristic
24	Q₈⋊(C₇⋊C₈)	C₂²	1	8	yes	—
25	#89253	C₂²	1	4	yes	—
26	C₈×C₇⋊C₈	C₂²	1	2	yes	—
27	C₈×C₇⋊C₈	C₂²	1	2	yes	—
28	#17958	—	7	1	no	maximal, Sylow
29	C₂×C₇×Q₁₆	C₂×C₄	1	4	yes	—
30	C₂²×C₇×C₈	C₂×C₄	1	2	yes	—
31	C₇×C₄⋊Q₈	C₂×C₄	1	2	yes	—
32	C₇×C₄⋊Q₈	C₂×C₄	1	2	yes	—
33	C₇×#125	—	2	2	no	—
34	C₇×C₂²⋊Q₈	—	2	2	no	—
35	C₇×C₂²⋊Q₈	C₂×C₄	1	4	yes	—
36	C₄×C₇×Q₈	—	2	2	no	—
37	C₄×C₇×Q₈	C₂³	1	4	yes	—
38	C₇×#117	—	2	2	no	—
39	C₂×C₄²×C₇	C₂³	1	1	yes	characteristic
40	C₂×#1136	—	2	2	no	—
41	C₂²×C₇⋊C₈	—	2	1	no	—
42	C₂²×C₇⋊C₈	D₄	1	1	yes	characteristic
43	C₂²×C₇⋊C₈	D₄	1	1	yes	characteristic
44	C₇×#107	C₂×C₄	1	4	yes	—
45	C₇×#103	C₂×C₄	1	8	yes	—
46	C₄×C₇×C₈	C₂×C₄	1	1	yes	characteristic
47	C₄×C₇×C₈	C₂×C₄	1	1	yes	characteristic
48	C₄×C₇×C₈	C₂³	1	2	yes	—
49	C₄⋊(C₇⋊C₈)	—	2	2	no	—
50	C₄⋊(C₇⋊C₈)	—	2	4	no	—
51	C₄⋊(C₇⋊C₈)	—	2	4	no	—
52	C₄⋊(C₇⋊C₈)	C₂³	1	4	yes	—
53	#6377	—	2	4	no	—
54	C₄×C₇⋊C₈	—	2	1	no	—
55	C₄×C₇⋊C₈	—	2	4	no	—
56	C₄×C₇⋊C₈	D₄	1	1	yes	characteristic
57	C₄×C₇⋊C₈	D₄	1	1	yes	characteristic
58	C₄×C₇⋊C₈	C₂³	1	2	yes	—
59	C₄×C₇⋊C₈	C₂³	1	1	yes	characteristic
60	C₄×C₇⋊C₈	C₂³	1	1	yes	characteristic
61	#3170	D₇	1	1	yes	characteristic
62	#2960	—	7	2	no	—
63	#1591	—	7	4	no	—
64	C₈×Q₁₆	—	7	4	no	—
65	#1567	—	7	1	no	—
66	#1493	—	7	2	no	—
67	C₂×C₈²	—	7	1	no	—
68	C₂×C₇×Q₈	—	2	2	no	—
69	C₂×C₇×Q₈	C₂²×C₄	1	4	yes	—
70	C₂²×C₄×C₇	C₂²×C₄	1	1	yes	characteristic
71	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
72	C₂²×C₄×C₇	#41	1	1	yes	characteristic
73	C₇×Q₁₆	—	2	8	no	—
74	C₂×C₇×C₈	—	2	4	no	—
75	C₂×C₇×C₈	C₂²×C₄	1	2	yes	—
76	C₂×C₇×C₈	C₂²×C₄	1	2	yes	—
77	C₇×C₄⋊C₄	—	2	2	no	—
78	C₇×C₄⋊C₄	—	2	2	no	—
79	C₇×C₄⋊C₄	—	2	2	no	—
80	C₇×C₄⋊C₄	—	2	4	no	—
81	C₇×C₄⋊C₄	—	2	4	no	—
82	C₇×C₄⋊C₄	C₂²×C₄	1	4	yes	—
83	C₇×C₂²⋊C₄	—	2	4	no	—
84	C₄²×C₇	—	2	4	no	—
85	C₄²×C₇	C₂²×C₄	1	1	yes	characteristic
86	C₄²×C₇	C₂×D₄	1	1	yes	characteristic
87	C₄²×C₇	#41	1	1	yes	characteristic
88	C₄²×C₇	C₂⁴	1	1	yes	characteristic
89	#1136	—	4	4	no	—
90	C₂×C₇⋊C₈	—	2	4	no	—
91	C₂×C₇⋊C₈	—	2	2	no	—
92	C₂×C₇⋊C₈	—	2	2	no	—
93	C₂×C₇⋊C₈	—	2	4	no	—
94	C₂×C₇⋊C₈	Q₁₆	1	4	yes	complemented
95	C₂×C₇⋊C₈	C₂×D₄	1	1	yes	characteristic
96	C₂×C₇⋊C₈	C₂×D₄	1	1	yes	characteristic
97	C₂×C₇⋊C₈	C₂×D₄	1	1	yes	characteristic
98	C₂×C₇⋊C₈	C₂×D₄	1	1	yes	characteristic
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	4	yes	—
104	#446	—	7	4	no	—
105	#446	—	14	2	no	—
106	C₈×Q₈	—	7	4	no	—
107	C₈×Q₈	—	14	2	no	—
108	C₄×Q₁₆	C₂×D₇	1	4	yes	—
109	#424	—	14	2	no	—
110	#423	—	7	2	no	—
111	#405	—	14	2	no	—
112	C₂×C₄×C₈	—	7	1	no	—
113	C₂×C₄×C₈	—	7	1	no	—
114	C₂×C₄×C₈	C₂×D₇	1	1	yes	characteristic
115	#335	—	7	4	no	—
116	Q₈⋊C₈	—	7	8	no	—
117	C₈²	—	7	2	no	—
118	C₈²	—	7	2	no	—
119	C₂³×C₇	C₄×D₄	1	1	yes	characteristic
120	C₇×Q₈	—	2	8	no	—
121	C₇×Q₈	—	4	4	no	—
122	C₂×C₄×C₇	—	2	2	no	—
123	C₂×C₄×C₇	—	2	2	no	—
124	C₂×C₄×C₇	—	2	4	no	—
125	C₂×C₄×C₇	—	2	4	no	—
126	C₂×C₄×C₇	C₄×D₄	1	1	yes	characteristic
127	C₂×C₄×C₇	C₄×D₄	1	1	yes	characteristic
128	C₂×C₄×C₇	C₄×D₄	1	1	yes	characteristic
129	C₂×C₄×C₇	C₂×Q₁₆	1	2	yes	—
130	C₂×C₄×C₇	#135	1	2	yes	—
131	C₂×C₄×C₇	C₂³×C₄	1	1	yes	characteristic
132	C₂×C₄×C₇	C₂²×D₄	1	1	yes	characteristic
133	C₂×C₄×C₇	C₂×#41	1	1	yes	characteristic
134	C₇×C₈	—	2	2	no	—
135	C₇×C₈	#131	1	4	yes	—
136	C₇⋊C₈	—	2	4	no	—
137	C₇⋊C₈	—	2	1	no	—
138	C₇⋊C₈	—	2	1	no	—
139	C₇⋊C₈	—	4	4	no	—
140	C₇⋊C₈	C₂×Q₁₆	1	2	yes	—
141	C₇⋊C₈	C₂×Q₁₆	1	2	yes	—
142	C₂×Q₁₆	C₂×Dic₇	1	4	yes	—
143	C₂×M₁₆	—	14	2	no	—
144	C₂²×C₈	—	7	1	no	—
145	C₂²×C₈	—	7	1	no	—
146	C₂²×C₈	—	14	1	no	—
147	C₂²×C₈	C₂×Dic₇	1	2	yes	—
148	C₄⋊Q₈	C₂×Dic₇	1	2	yes	—
149	C₄⋊Q₈	C₂×Dic₇	1	2	yes	—
150	#125	—	2	2	no	—
151	C₂²⋊Q₈	—	2	2	no	—
152	C₂²⋊Q₈	C₂×Dic₇	1	4	yes	—
153	C₄×Q₈	—	2	2	no	—
154	C₄×Q₈	C₂²×D₇	1	4	yes	—
155	#117	—	2	2	no	—
156	C₂×C₄²	C₂²×D₇	1	1	yes	characteristic
157	#107	C₂×Dic₇	1	4	yes	—
158	C₄⋊C₈	—	7	4	no	—
159	C₄⋊C₈	—	14	4	no	—
160	C₄⋊C₈	—	14	4	no	—
161	C₄⋊C₈	—	14	2	no	—
162	#103	C₂×Dic₇	1	8	yes	—
163	#97	—	14	4	no	—
164	C₄×C₈	—	7	1	no	—
165	C₄×C₈	—	7	1	no	—
166	C₄×C₈	—	7	1	no	—
167	C₄×C₈	—	7	1	no	—
168	C₄×C₈	—	7	2	no	—
169	C₄×C₈	—	14	4	no	—
170	C₄×C₈	—	14	1	no	—
171	C₄×C₈	C₂×Dic₇	1	1	yes	characteristic
172	C₄×C₈	C₂×Dic₇	1	1	yes	characteristic
173	C₄×C₈	C₂²×D₇	1	2	yes	—
174	C₂²×C₇	—	2	1	no	—
175	C₂²×C₇	—	2	1	no	—
176	C₂²×C₇	C₄×Q₁₆	1	2	yes	—
177	C₂²×C₇	C₂×C₄×D₄	1	1	yes	characteristic
178	C₄×C₇	—	2	1	no	—
179	C₄×C₇	—	2	1	no	—
180	C₄×C₇	—	4	4	no	—
181	C₄×C₇	—	4	4	no	—
182	C₄×C₇	C₄×Q₁₆	1	2	yes	—
183	C₄×C₇	#443	1	2	yes	—
184	C₄×C₇	C₂×#131	1	2	yes	—
185	C₄×C₇	C₂²×Q₁₆	1	1	yes	characteristic
186	C₄×C₇	C₂×#135	1	1	yes	characteristic
187	C₂×Q₈	—	2	2	no	—
188	C₂×Q₈	C₂²×Dic₇	1	4	yes	—
189	C₂²×C₄	D₄×D₇	1	1	yes	characteristic
190	C₂²×C₄	#1159	1	1	yes	characteristic
191	C₂²×C₄	C₂²×Dic₇	1	1	yes	characteristic
192	Q₁₆	—	2	8	no	—
193	M₁₆	—	28	4	no	—
194	C₂×C₈	—	2	4	no	—
195	C₂×C₈	—	7	4	no	—
196	C₂×C₈	—	7	1	no	—
197	C₂×C₈	—	7	1	no	—
198	C₂×C₈	—	7	1	no	—
199	C₂×C₈	—	7	1	no	—
200	C₂×C₈	—	14	4	no	—
201	C₂×C₈	—	14	2	no	—
202	C₂×C₈	—	14	2	no	—
203	C₂×C₈	—	14	4	no	—
204	C₂×C₈	C₂²×Dic₇	1	2	yes	—
205	C₂×C₈	C₂²×Dic₇	1	2	yes	—
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	—	2	2	no	—
219	C₂×C₇	#2931	1	1	yes	characteristic
220	C₂×C₇	C₂×C₄×Q₁₆	1	1	yes	characteristic
221	C₂×C₇	C₂×#443	1	1	yes	characteristic
222	C₂³	D₄×Dic₇	1	1	yes	characteristic
223	Q₈	—	2	8	no	—
224	Q₈	—	4	4	no	—
225	C₂×C₄	—	2	2	no	—
226	C₂×C₄	—	2	2	no	—
227	C₂×C₄	—	2	4	no	—
228	C₂×C₄	—	2	4	no	—
229	C₂×C₄	D₇×Q₁₆	1	2	yes	—
230	C₂×C₄	#6482	1	2	yes	—
231	C₂×C₄	D₄×Dic₇	1	1	yes	characteristic
232	C₂×C₄	D₄×Dic₇	1	1	yes	characteristic
233	C₂×C₄	D₄×Dic₇	1	1	yes	characteristic
234	C₂×C₄	C₂×D₄×D₇	1	1	yes	characteristic
235	C₂×C₄	C₂×#1159	1	1	yes	characteristic
236	C₂×C₄	C₂³×Dic₇	1	1	yes	characteristic
237	C₈	—	2	2	no	—
238	C₈	—	7	2	no	—
239	C₈	—	7	2	no	—
240	C₈	—	14	4	no	—
241	C₈	—	14	1	no	—
242	C₈	—	14	1	no	—
243	C₈	—	28	4	no	—
244	C₈	#6511	1	4	yes	—
245	C₇	#17958	1	1	yes	characteristic, complemented, Sylow
246	C₂²	—	2	1	no	—
247	C₂²	—	2	1	no	—
248	C₂²	Q₁₆×Dic₇	1	2	yes	—
249	C₂²	C₂×D₄×Dic₇	1	1	yes	characteristic
250	C₄	—	2	1	no	—
251	C₄	—	2	1	no	—
252	C₄	—	4	4	no	—
253	C₄	—	4	4	no	—
254	C₄	Q₁₆×Dic₇	1	2	yes	—
255	C₄	#89968	1	2	yes	—
256	C₄	C₂×D₇×Q₁₆	1	1	yes	characteristic
257	C₄	C₂×#6482	1	1	yes	characteristic
258	C₄	C₂×#6511	1	2	yes	—
259	C₂	—	2	2	no	—
260	C₂	#11740720	1	1	yes	characteristic
261	C₂	C₂×Q₁₆×Dic₇	1	1	yes	characteristic
262	C₂	C₂×#89968	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	#17958
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. #11728659	C₂	semidirect
5. Q₁₆×C₇⋊C₈	C₂	semidirect
94. C₂×C₇⋊C₈	Q₁₆	semidirect
245. C₇	#17958	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
#11740773	2
#11729580	2
#11728659	2
Q₁₆×C₇⋊C₈	2
#11727485	2
C₂×C₈×C₇⋊C₈	2
#17958	7