Subgroups — #13246830

Order 1792.

404 conjugacy classes of subgroups.

The subgroup lattice has 272 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₇×#2931	C₂	1	1	yes	maximal, characteristic
2	#11729564	C₂	1	2	yes	maximal
3	#11727485	C₂	1	1	yes	maximal, characteristic, complemented
4	#11727484	C₂	1	1	yes	maximal, characteristic, complemented
5	C₇⋊#1309	C₂	1	2	yes	maximal, complemented
6	C₂×C₇×#131	C₄	1	2	yes	—
7	C₇×#532	C₂²	1	1	yes	characteristic
8	C₇×#436	—	2	1	no	—
9	C₇×#436	—	2	1	no	—
10	C₇×C₈×D₄	—	2	1	no	—
11	C₇×C₈×D₄	—	2	1	no	—
12	C₇×#423	C₂²	1	1	yes	characteristic
13	C₇×#408	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	#89341	—	2	2	no	—
18	#89339	—	2	2	no	—
19	#89253	—	2	1	no	—
20	#89253	—	2	1	no	—
21	#89252	—	2	1	no	—
22	#89252	—	2	1	no	—
23	#8640	—	7	1	no	maximal, Sylow
24	C₂×C₇×#41	C₂×C₄	1	2	yes	—
25	C₇×#131	—	4	4	no	—
26	C₂×C₇×M₁₆	—	2	2	no	—
27	C₂×C₇×M₁₆	C₂×C₄	1	2	yes	—
28	C₂²×C₇×C₈	—	2	2	no	—
29	C₂²×C₇×C₈	C₂×C₄	1	1	yes	characteristic
30	C₂²×C₇×C₈	C₂×C₄	1	1	yes	characteristic
31	C₇×C₄⋊Q₈	D₄	1	1	yes	characteristic
32	C₇×#127	D₄	1	1	yes	characteristic
33	C₇×#124	Q₈	1	2	yes	—
34	C₇×#121	—	2	1	no	—
35	C₇×#121	—	2	1	no	—
36	C₄×C₇×D₄	—	2	1	no	—
37	C₄×C₇×D₄	—	2	1	no	—
38	C₂×C₄²×C₇	C₂³	1	1	yes	characteristic
39	C₂×#1136	—	2	2	no	—
40	C₂×#1136	—	2	2	no	—
41	C₇×C₄⋊C₈	—	2	1	no	—
42	C₇×C₄⋊C₈	—	2	1	no	—
43	C₇×C₂²⋊C₈	—	2	2	no	—
44	C₇×C₂²⋊C₈	—	2	2	no	—
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	D₄	1	1	yes	characteristic
49	C₄⋊(C₇⋊C₈)	—	2	2	no	—
50	C₄⋊(C₇⋊C₈)	—	2	2	no	—
51	C₄⋊(C₇⋊C₈)	—	2	2	no	—
52	C₄⋊(C₇⋊C₈)	—	2	2	no	—
53	#2931	D₇	1	1	yes	characteristic
54	#1567	—	7	1	no	—
55	#1566	—	7	1	no	—
56	#1560	—	7	2	no	—
57	#1309	—	7	2	no	—
58	C₇×#41	—	4	2	no	—
59	C₇×#41	—	4	2	no	—
60	C₂×C₇×Q₈	C₄⋊C₄	1	2	yes	—
61	C₂×C₇×D₄	—	2	2	no	—
62	C₂×C₇×D₄	—	2	1	no	—
63	C₂×C₇×D₄	—	2	1	no	—
64	C₂×C₇×D₄	C₄⋊C₄	1	2	yes	—
65	C₂²×C₄×C₇	—	2	2	no	—
66	C₂²×C₄×C₇	C₂²×C₄	1	1	yes	characteristic
67	C₂²×C₄×C₇	#41	1	1	yes	characteristic
68	C₂²×C₄×C₇	#41	1	1	yes	characteristic
69	C₇×M₁₆	—	4	4	no	—
70	C₇×M₁₆	—	4	2	no	—
71	C₂×C₇×C₈	—	2	2	no	—
72	C₂×C₇×C₈	—	2	2	no	—
73	C₂×C₇×C₈	—	4	1	no	—
74	C₂×C₇×C₈	—	4	4	no	—
75	C₂×C₇×C₈	C₂²⋊C₄	1	1	yes	characteristic
76	C₂×C₇×C₈	C₂²⋊C₄	1	1	yes	characteristic
77	C₂×C₇×C₈	C₂²⋊C₄	1	1	yes	characteristic
78	C₂×C₇×C₈	C₂²⋊C₄	1	1	yes	characteristic
79	C₇×C₄⋊C₄	—	2	1	no	—
80	C₇×C₄⋊C₄	—	2	1	no	—
81	C₇×C₂²⋊C₄	—	2	2	no	—
82	C₇×C₂²⋊C₄	—	2	2	no	—
83	C₄²×C₇	C₂×D₄	1	1	yes	characteristic
84	C₄²×C₇	C₂×D₄	1	1	yes	characteristic
85	C₄²×C₇	C₂×D₄	1	1	yes	characteristic
86	C₄²×C₇	C₂×Q₈	1	1	yes	characteristic
87	#1136	—	8	2	no	—
88	#1136	—	8	2	no	—
89	C₂×C₇⋊C₈	—	4	2	no	—
90	C₂×C₇⋊C₈	—	4	2	no	—
91	C₂×#131	Dic₇	1	2	yes	—
92	#532	C₂×D₇	1	1	yes	characteristic
93	#436	—	2	1	no	—
94	#436	—	2	1	no	—
95	C₈×D₄	—	2	1	no	—
96	C₈×D₄	—	2	1	no	—
97	#423	—	7	2	no	—
98	#423	—	7	2	no	—
99	#423	C₂×D₇	1	1	yes	characteristic
100	#408	Dic₇	1	2	yes	—
101	C₂×C₄×C₈	C₂×D₇	1	1	yes	characteristic
102	#335	—	14	1	no	—
103	#335	—	14	1	no	—
104	#334	—	14	1	no	—
105	#334	—	14	1	no	—
106	#333	—	14	2	no	—
107	#331	—	14	2	no	—
108	C₂³×C₇	—	2	2	no	—
109	C₂³×C₇	#117	1	1	yes	characteristic
110	C₇×Q₈	—	4	2	no	—
111	C₇×D₄	—	4	2	no	—
112	C₇×D₄	—	4	4	no	—
113	C₇×D₄	—	4	2	no	—
114	C₇×D₄	—	4	1	no	—
115	C₇×D₄	—	4	1	no	—
116	C₂×C₄×C₇	—	2	1	no	—
117	C₂×C₄×C₇	—	2	1	no	—
118	C₂×C₄×C₇	—	2	2	no	—
119	C₂×C₄×C₇	—	2	1	no	—
120	C₂×C₄×C₇	—	2	1	no	—
121	C₂×C₄×C₇	—	2	2	no	—
122	C₂×C₄×C₇	—	4	2	no	—
123	C₂×C₄×C₇	—	4	2	no	—
124	C₂×C₄×C₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
125	C₂×C₄×C₇	C₂×C₄⋊C₄	1	1	yes	characteristic
126	C₂×C₄×C₇	#121	1	1	yes	characteristic
127	C₂×C₄×C₇	#121	1	1	yes	characteristic
128	C₂×C₄×C₇	C₂²⋊Q₈	1	1	yes	characteristic
129	C₂×C₄×C₇	C₂²⋊Q₈	1	1	yes	characteristic
130	C₇×C₈	—	2	1	no	—
131	C₇×C₈	—	2	1	no	—
132	C₇×C₈	—	2	1	no	—
133	C₇×C₈	—	4	2	no	—
134	C₇×C₈	#100	1	2	yes	—
135	C₇⋊C₈	—	8	2	no	—
136	C₇⋊C₈	—	8	2	no	—
137	C₂×#41	C₂×Dic₇	1	2	yes	—
138	#131	—	4	4	no	—
139	C₂×M₁₆	—	2	2	no	—
140	C₂×M₁₆	—	14	2	no	—
141	C₂×M₁₆	—	14	2	no	—
142	C₂×M₁₆	C₂×Dic₇	1	2	yes	—
143	C₂²×C₈	—	2	2	no	—
144	C₂²×C₈	C₂×Dic₇	1	1	yes	characteristic
145	C₂²×C₈	C₂×Dic₇	1	1	yes	characteristic
146	C₄⋊Q₈	D₂₈	1	1	yes	characteristic
147	#127	D₂₈	1	1	yes	characteristic
148	#124	Dic₁₄	1	2	yes	—
149	#121	—	2	1	no	—
150	#121	—	2	1	no	—
151	C₄×D₄	—	2	1	no	—
152	C₄×D₄	—	2	1	no	—
153	C₂×C₄²	C₂²×D₇	1	1	yes	characteristic
154	C₄⋊C₈	—	2	1	no	—
155	C₄⋊C₈	—	2	1	no	—
156	C₄⋊C₈	—	14	2	no	—
157	C₄⋊C₈	—	14	2	no	—
158	C₄⋊C₈	—	14	2	no	—
159	C₄⋊C₈	—	14	2	no	—
160	C₂²⋊C₈	—	2	2	no	—
161	C₂²⋊C₈	—	2	2	no	—
162	C₄×C₈	C₂²⋊D₇	1	1	yes	characteristic
163	C₄×C₈	C₂²⋊D₇	1	1	yes	characteristic
164	C₄×C₈	C₂²⋊D₇	1	1	yes	characteristic
165	C₄×C₈	C₂²⋊D₇	1	1	yes	characteristic
166	C₂²×C₇	—	2	1	no	—
167	C₂²×C₇	—	2	1	no	—
168	C₂²×C₇	—	2	1	no	—
169	C₂²×C₇	—	4	2	no	—
170	C₂²×C₇	—	4	2	no	—
171	C₂²×C₇	—	4	2	no	—
172	C₂²×C₇	#380	1	1	yes	characteristic
173	C₄×C₇	—	2	1	no	—
174	C₄×C₇	—	2	1	no	—
175	C₄×C₇	—	2	1	no	—
176	C₄×C₇	—	4	2	no	—
177	C₄×C₇	C₂×#100	1	1	yes	characteristic
178	C₄×C₇	#413	1	1	yes	characteristic
179	C₄×C₇	#471	1	2	yes	—
180	C₄×C₇	#472	1	1	yes	characteristic
181	C₄×C₇	#473	1	1	yes	characteristic
182	#41	—	4	2	no	—
183	#41	—	4	2	no	—
184	C₂×Q₈	C₄⋊Dic₇	1	2	yes	—
185	C₂×D₄	—	2	2	no	—
186	C₂×D₄	—	2	1	no	—
187	C₂×D₄	—	2	1	no	—
188	C₂×D₄	C₄⋊Dic₇	1	2	yes	—
189	C₂²×C₄	—	2	2	no	—
190	C₂²×C₄	#1157	1	1	yes	characteristic
191	C₂²×C₄	#1157	1	1	yes	characteristic
192	C₂²×C₄	C₂²×Dic₇	1	1	yes	characteristic
193	M₁₆	—	4	4	no	—
194	M₁₆	—	4	2	no	—
195	M₁₆	—	56	2	no	—
196	M₁₆	—	56	2	no	—
197	C₂×C₈	—	2	2	no	—
198	C₂×C₈	—	2	2	no	—
199	C₂×C₈	—	4	1	no	—
200	C₂×C₈	—	4	4	no	—
201	C₂×C₈	—	28	2	no	—
202	C₂×C₈	—	28	2	no	—
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	C₂×C₈	C₂²⋊Dic₇	1	1	yes	characteristic
207	C₄⋊C₄	—	2	1	no	—
208	C₄⋊C₄	—	2	1	no	—
209	C₂²⋊C₄	—	2	2	no	—
210	C₂²⋊C₄	—	2	2	no	—
211	C₄²	C₂×Dic₁₄	1	1	yes	characteristic
212	C₄²	C₂×D₂₈	1	1	yes	characteristic
213	C₄²	C₂×C₂²⋊D₇	1	1	yes	characteristic
214	C₄²	C₂×C₂²⋊D₇	1	1	yes	characteristic
215	C₂×C₇	—	4	1	no	—
216	C₂×C₇	—	8	2	no	—
217	C₂×C₇	#1800	1	1	yes	characteristic
218	C₂×C₇	#1815	1	1	yes	characteristic
219	C₂×C₇	#1816	1	1	yes	characteristic
220	C₂³	—	2	2	no	—
221	C₂³	#6489	1	1	yes	characteristic
222	Q₈	—	4	2	no	—
223	D₄	—	4	2	no	—
224	D₄	—	4	4	no	—
225	D₄	—	4	2	no	—
226	D₄	—	4	1	no	—
227	D₄	—	4	1	no	—
228	C₂×C₄	—	2	2	no	—
229	C₂×C₄	—	2	1	no	—
230	C₂×C₄	—	2	1	no	—
231	C₂×C₄	—	2	1	no	—
232	C₂×C₄	—	2	1	no	—
233	C₂×C₄	—	2	2	no	—
234	C₂×C₄	—	4	2	no	—
235	C₂×C₄	—	4	2	no	—
236	C₂×C₄	#6487	1	1	yes	characteristic
237	C₂×C₄	#6487	1	1	yes	characteristic
238	C₂×C₄	C₂×C₄⋊Dic₇	1	1	yes	characteristic
239	C₂×C₄	#6493	1	1	yes	characteristic
240	C₂×C₄	#6493	1	1	yes	characteristic
241	C₂×C₄	C₂×C₂²⋊Dic₇	1	1	yes	characteristic
242	C₈	—	2	1	no	—
243	C₈	—	2	1	no	—
244	C₈	—	2	1	no	—
245	C₈	—	4	2	no	—
246	C₈	—	56	2	no	—
247	C₈	—	56	2	no	—
248	C₈	#6407	1	2	yes	—
249	C₇	#8640	1	1	yes	characteristic, complemented, Sylow
250	C₂²	—	2	1	no	—
251	C₂²	—	2	1	no	—
252	C₂²	—	2	1	no	—
253	C₂²	—	4	2	no	—
254	C₂²	—	4	2	no	—
255	C₂²	—	4	2	no	—
256	C₂²	#89984	1	1	yes	characteristic
257	C₄	—	2	1	no	—
258	C₄	—	2	1	no	—
259	C₄	—	2	1	no	—
260	C₄	—	4	2	no	—
261	C₄	#89913	1	2	yes	—
262	C₄	#89914	1	1	yes	characteristic
263	C₄	#89915	1	1	yes	characteristic
264	C₄	C₂×#6407	1	1	yes	characteristic
265	C₄	#90009	1	1	yes	characteristic
266	C₂	—	4	1	no	—
267	C₂	—	8	2	no	—
268	C₂	#11729371	1	1	yes	characteristic
269	C₂	#11729769	1	1	yes	characteristic
270	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×#2931	C₂
socle	C₂²×C₇	#380
prime radical	C₇×#532	C₂²
cosocle	C₂×C₄²×C₇	C₂³
hypercenter	#2931	D₇
hypocenter	C₇	#8640
LCS	G ▷ C₂²×C₄×C₇ ▷ C₂²×C₇ ▷ C₇
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #2931

Complements

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

N	H ≅ G/N	type
3. #11727485	C₂	semidirect
4. #11727484	C₂	semidirect
5. C₇⋊#1309	C₂	semidirect
249. C₇	#8640	semidirect

Maximal subgroups

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

subgroup	index
C₇×#2931	2
#11729564	2
#11727485	2
#11727484	2
C₇⋊#1309	2
#8640	7