Subgroups — #13509321

Order 1792.

514 conjugacy classes of subgroups.

The subgroup lattice has 274 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₇×#3160	C₂	1	1	yes	maximal, characteristic
2	C₂×#89454	C₂	1	1	yes	maximal, characteristic
3	#11729139	C₂	1	2	yes	maximal, complemented
4	#11729138	C₂	1	2	yes	maximal, complemented
5	#11729109	C₂	1	4	yes	maximal, complemented
6	#11728403	C₂	1	4	yes	maximal
7	C₄×#6386	C₂	1	1	yes	maximal, characteristic
8	C₇×#533	—	2	1	no	—
9	C₂×#6432	—	2	1	no	—
10	C₇×#500	C₂²	1	1	yes	characteristic
11	C₇×#500	C₂²	1	1	yes	characteristic
12	C₇×#499	—	2	1	no	—
13	C₇×#499	C₂²	1	2	yes	—
14	C₇×#498	—	2	1	no	—
15	C₇×#427	—	2	1	no	—
16	C₇×#427	C₂²	1	2	yes	—
17	C₂×C₄×C₇×C₈	C₂²	1	1	yes	characteristic
18	C₂×#6392	—	2	1	no	—
19	C₂×#6392	C₂²	1	2	yes	—
20	C₂×#6386	C₂²	1	2	yes	—
21	#89454	—	2	1	no	—
22	#89454	—	2	1	no	—
23	#89454	C₂²	1	2	yes	—
24	#89454	C₂²	1	2	yes	—
25	#89285	C₂²	1	8	yes	—
26	#89284	C₂²	1	8	yes	—
27	#89256	C₂²	1	2	yes	—
28	C₄×C₇⋊C₁₆	C₂²	1	2	yes	—
29	#33731	—	7	1	no	maximal, Sylow
30	C₂²×Dic₁₄	—	4	1	no	—
31	C₂²×C₇×C₈	D₄	1	1	yes	characteristic
32	C₂²×C₇×C₈	C₂³	1	1	yes	characteristic
33	C₇×C₄⋊Q₈	—	2	1	no	—
34	C₇×C₄⋊Q₈	—	2	1	no	—
35	C₇×#125	—	2	2	no	—
36	C₇×C₂²⋊Q₈	—	4	1	no	—
37	C₇×C₂²⋊Q₈	—	4	1	no	—
38	C₄×C₇×Q₈	—	4	1	no	—
39	C₄×C₇×Q₈	—	4	1	no	—
40	C₇×#117	—	2	2	no	—
41	C₂×C₄²×C₇	D₄	1	1	yes	characteristic
42	C₂×C₄⋊Dic₇	—	2	2	no	—
43	C₂×C₄⋊Dic₇	—	4	1	no	—
44	#6432	—	2	2	no	—
45	#6432	—	2	2	no	—
46	#6432	—	4	1	no	—
47	#6432	—	4	1	no	—
48	C₇×#107	—	2	1	no	—
49	C₇×#107	—	2	1	no	—
50	C₇×#107	C₂³	1	2	yes	—
51	C₇×#107	C₂³	1	2	yes	—
52	C₇×#106	—	2	2	no	—
53	C₇×#106	—	2	2	no	—
54	C₄×C₇×C₈	D₄	1	1	yes	characteristic
55	C₄×C₇×C₈	D₄	1	1	yes	characteristic
56	C₄×C₇×C₈	C₂³	1	1	yes	characteristic
57	C₄×C₇×C₈	C₂³	1	1	yes	characteristic
58	#6392	—	2	2	no	—
59	#6392	—	2	2	no	—
60	#6392	—	2	2	no	—
61	#6392	C₂³	1	4	yes	—
62	#6386	Q₈	1	8	yes	complemented
63	C₂×C₇⋊C₁₆	C₂³	1	4	yes	—
64	#3160	D₇	1	1	yes	characteristic
65	C₂×#500	—	7	1	no	—
66	#2255	—	7	4	no	—
67	#2254	—	7	2	no	—
68	#2252	—	7	2	no	—
69	#2159	—	7	4	no	—
70	C₄×M₃₂	—	7	1	no	—
71	C₂×C₇×Q₈	—	4	1	no	—
72	C₂×C₇×Q₈	—	4	1	no	—
73	C₂²×C₄×C₇	D₈	1	1	yes	characteristic
74	C₂²×C₄×C₇	D₈	1	1	yes	characteristic
75	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
76	C₂²×Dic₇	—	4	2	no	—
77	C₂×Dic₁₄	—	4	2	no	—
78	C₂×Dic₁₄	—	4	2	no	—
79	C₂×Dic₁₄	—	8	4	no	—
80	C₂×C₇×C₈	Q₁₆	1	4	yes	—
81	C₂×C₇×C₈	C₂×D₄	1	1	yes	characteristic
82	C₂×C₇×C₈	C₂×D₄	1	1	yes	characteristic
83	C₂×C₇×C₈	C₂×D₄	1	1	yes	characteristic
84	C₂×C₇×C₈	C₂×Q₈	1	4	yes	—
85	C₂×C₇×C₈	C₂⁴	1	1	yes	characteristic
86	C₇×C₄⋊C₄	—	2	2	no	—
87	C₇×C₄⋊C₄	—	2	2	no	—
88	C₇×C₄⋊C₄	—	4	2	no	—
89	C₇×C₄⋊C₄	—	4	2	no	—
90	C₇×C₄⋊C₄	—	4	1	no	—
91	C₇×C₄⋊C₄	—	4	1	no	—
92	C₇×C₂²⋊C₄	—	4	2	no	—
93	C₄²×C₇	—	4	2	no	—
94	C₄²×C₇	D₈	1	1	yes	characteristic
95	C₄²×C₇	D₈	1	1	yes	characteristic
96	C₄²×C₇	C₂×D₄	1	1	yes	characteristic
97	C₄²×C₇	C₂×D₄	1	1	yes	characteristic
98	C₄⋊Dic₇	—	2	4	no	—
99	C₄⋊Dic₇	—	4	2	no	—
100	C₄⋊Dic₇	—	4	2	no	—
101	C₄⋊Dic₇	—	4	2	no	—
102	C₇⋊C₁₆	C₂×Q₈	1	8	yes	complemented
103	#533	—	2	1	no	—
104	C₂×C₄⋊Q₈	—	14	1	no	—
105	C₂×M₃₂	—	7	2	no	—
106	#500	—	7	2	no	—
107	#500	—	7	2	no	—
108	#500	—	14	1	no	—
109	#500	—	14	1	no	—
110	#500	C₂×D₇	1	1	yes	characteristic
111	#500	C₂×D₇	1	1	yes	characteristic
112	#499	—	2	1	no	—
113	#499	C₂×D₇	1	2	yes	—
114	#498	—	2	1	no	—
115	#427	—	2	1	no	—
116	#427	C₂×D₇	1	2	yes	—
117	C₂×#107	—	7	2	no	—
118	C₂×#107	—	14	1	no	—
119	C₂×C₄×C₈	C₂×D₇	1	1	yes	characteristic
120	#367	—	7	8	no	—
121	#366	—	7	8	no	—
122	#346	—	7	2	no	—
123	C₄×C₁₆	—	7	2	no	—
124	C₂³×C₇	C₂×D₈	1	1	yes	characteristic
125	C₇×Q₈	—	8	2	no	—
126	C₇×Q₈	—	8	2	no	—
127	C₂×C₄×C₇	—	2	2	no	—
128	C₂×C₄×C₇	—	2	2	no	—
129	C₂×C₄×C₇	—	4	2	no	—
130	C₂×C₄×C₇	—	4	2	no	—
131	C₂×C₄×C₇	C₄⋊Q₈	1	2	yes	—
132	C₂×C₄×C₇	C₂×D₈	1	1	yes	characteristic
133	C₂×C₄×C₇	C₂×D₈	1	1	yes	characteristic
134	C₂×C₄×C₇	C₂×D₈	1	1	yes	characteristic
135	C₂×C₄×C₇	C₂×D₈	1	1	yes	characteristic
136	C₂×C₄×C₇	C₂×D₈	1	1	yes	characteristic
137	C₂×C₄×C₇	C₂×Q₁₆	1	2	yes	—
138	C₂×C₄×C₇	C₂²×D₄	1	1	yes	characteristic
139	C₂×Dic₇	—	4	4	no	—
140	C₂×Dic₇	—	8	4	no	—
141	Dic₁₄	—	8	4	no	—
142	Dic₁₄	—	8	4	no	—
143	C₇×C₈	C₄⋊Q₈	1	2	yes	—
144	C₇×C₈	C₂×Q₁₆	1	2	yes	—
145	C₇×C₈	C₂×Q₁₆	1	2	yes	—
146	C₇×C₈	C₂²×Q₈	1	2	yes	—
147	C₂²×Q₈	—	28	1	no	—
148	C₂²×C₈	C₂²⋊D₇	1	1	yes	characteristic
149	C₂²×C₈	C₂²×D₇	1	1	yes	characteristic
150	C₄⋊Q₈	—	2	1	no	—
151	C₄⋊Q₈	—	2	1	no	—
152	C₄⋊Q₈	—	14	2	no	—
153	C₄⋊Q₈	—	14	2	no	—
154	C₄⋊Q₈	—	28	1	no	—
155	C₄⋊Q₈	—	28	1	no	—
156	#125	—	2	2	no	—
157	C₂²⋊Q₈	—	4	1	no	—
158	C₂²⋊Q₈	—	4	1	no	—
159	C₄×Q₈	—	4	1	no	—
160	C₄×Q₈	—	4	1	no	—
161	#117	—	2	2	no	—
162	C₂×C₄⋊C₄	—	14	2	no	—
163	C₂×C₄⋊C₄	—	28	1	no	—
164	C₂×C₄²	C₂²⋊D₇	1	1	yes	characteristic
165	M₃₂	—	7	8	no	—
166	C₂×C₁₆	—	7	4	no	—
167	#107	—	2	1	no	—
168	#107	—	2	1	no	—
169	#107	—	7	4	no	—
170	#107	—	14	2	no	—
171	#107	—	14	2	no	—
172	#107	—	14	2	no	—
173	#107	C₂²×D₇	1	2	yes	—
174	#107	C₂²×D₇	1	2	yes	—
175	#106	—	2	2	no	—
176	#106	—	2	2	no	—
177	C₄×C₈	C₂²⋊D₇	1	1	yes	characteristic
178	C₄×C₈	C₂²⋊D₇	1	1	yes	characteristic
179	C₄×C₈	C₂²×D₇	1	1	yes	characteristic
180	C₄×C₈	C₂²×D₇	1	1	yes	characteristic
181	C₂²×C₇	—	2	1	no	—
182	C₂²×C₇	—	2	1	no	—
183	C₂²×C₇	#500	1	2	yes	—
184	C₂²×C₇	C₂²×D₈	1	1	yes	characteristic
185	C₄×C₇	—	2	1	no	—
186	C₄×C₇	—	2	1	no	—
187	C₄×C₇	—	8	2	no	—
188	C₄×C₇	—	8	2	no	—
189	C₄×C₇	#500	1	2	yes	—
190	C₄×C₇	#509	1	2	yes	—
191	C₄×C₇	#510	1	2	yes	—
192	C₄×C₇	C₂×C₄⋊Q₈	1	1	yes	characteristic
193	C₄×C₇	C₂²×Q₁₆	1	1	yes	characteristic
194	Dic₇	—	8	4	no	—
195	C₂×Q₈	—	4	1	no	—
196	C₂×Q₈	—	4	1	no	—
197	C₂×Q₈	—	28	2	no	—
198	C₂×Q₈	—	28	2	no	—
199	C₂×Q₈	—	56	4	no	—
200	C₂²×C₄	—	28	2	no	—
201	C₂²×C₄	#1141	1	1	yes	characteristic
202	C₂²×C₄	#1141	1	1	yes	characteristic
203	C₂²×C₄	C₂×C₂²⋊D₇	1	1	yes	characteristic
204	C₂×C₈	#1144	1	4	yes	—
205	C₂×C₈	Q₈×D₇	1	4	yes	—
206	C₂×C₈	C₂×C₂²⋊D₇	1	1	yes	characteristic
207	C₂×C₈	C₂×C₂²⋊D₇	1	1	yes	characteristic
208	C₂×C₈	C₂×C₂²⋊D₇	1	1	yes	characteristic
209	C₂×C₈	C₂³×D₇	1	1	yes	characteristic
210	C₄⋊C₄	—	2	2	no	—
211	C₄⋊C₄	—	2	2	no	—
212	C₄⋊C₄	—	4	2	no	—
213	C₄⋊C₄	—	4	2	no	—
214	C₄⋊C₄	—	4	1	no	—
215	C₄⋊C₄	—	4	1	no	—
216	C₄⋊C₄	—	14	4	no	—
217	C₄⋊C₄	—	28	2	no	—
218	C₄⋊C₄	—	28	2	no	—
219	C₄⋊C₄	—	28	2	no	—
220	C₂²⋊C₄	—	4	2	no	—
221	C₄²	—	4	2	no	—
222	C₄²	#1141	1	1	yes	characteristic
223	C₄²	#1141	1	1	yes	characteristic
224	C₄²	C₂×C₂²⋊D₇	1	1	yes	characteristic
225	C₄²	C₂×C₂²⋊D₇	1	1	yes	characteristic
226	C₁₆	—	7	8	no	—
227	C₂×C₇	—	2	2	no	—
228	C₂×C₇	C₂×#500	1	1	yes	characteristic
229	C₂×C₇	C₂×#509	1	1	yes	characteristic
230	C₂×C₇	C₂×#510	1	1	yes	characteristic
231	C₂³	C₂×#1141	1	1	yes	characteristic
232	Q₈	—	8	2	no	—
233	Q₈	—	8	2	no	—
234	Q₈	—	56	4	no	—
235	Q₈	—	56	4	no	—
236	C₂×C₄	—	2	2	no	—
237	C₂×C₄	—	2	2	no	—
238	C₂×C₄	—	4	2	no	—
239	C₂×C₄	—	4	2	no	—
240	C₂×C₄	—	28	4	no	—
241	C₂×C₄	—	56	4	no	—
242	C₂×C₄	C₂×#1141	1	1	yes	characteristic
243	C₂×C₄	C₂×#1141	1	1	yes	characteristic
244	C₂×C₄	C₂×#1141	1	1	yes	characteristic
245	C₂×C₄	C₂×#1141	1	1	yes	characteristic
246	C₂×C₄	C₂×#1141	1	1	yes	characteristic
247	C₂×C₄	C₂×#1144	1	2	yes	—
248	C₂×C₄	#6507	1	2	yes	—
249	C₂×C₄	C₂²×C₂²⋊D₇	1	1	yes	characteristic
250	C₈	C₂×#1144	1	2	yes	—
251	C₈	C₂×#1144	1	2	yes	—
252	C₈	#6507	1	2	yes	—
253	C₈	C₂×Q₈×D₇	1	2	yes	—
254	C₇	#33731	1	1	yes	characteristic, complemented, Sylow
255	C₂²	—	2	1	no	—
256	C₂²	—	2	1	no	—
257	C₂²	#89850	1	2	yes	—
258	C₂²	C₂²×#1141	1	1	yes	characteristic
259	C₄	—	2	1	no	—
260	C₄	—	2	1	no	—
261	C₄	—	8	2	no	—
262	C₄	—	8	2	no	—
263	C₄	—	56	4	no	—
264	C₄	#89850	1	2	yes	—
265	C₄	#89965	1	2	yes	—
266	C₄	#89967	1	2	yes	—
267	C₄	C₂²×#1144	1	1	yes	characteristic
268	C₄	C₂×#6507	1	1	yes	characteristic
269	C₂	—	2	2	no	—
270	C₂	C₂×#89850	1	1	yes	characteristic
271	C₂	C₂×#89965	1	1	yes	characteristic
272	C₂	C₂×#89967	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂²×#1141
commutator, cosocle	C₂×C₇×C₈	C₂⁴
Frattini	C₂×C₈	C₂³×D₇
Fitting	C₇×#3160	C₂
socle	C₂²×C₇	C₂²×D₈
prime radical	C₂³×C₇	C₂×D₈
hypercenter	#3160	D₇
hypocenter	C₇	#33731
LCS	G ▷ C₂×C₇×C₈ ▷ C₄×C₇ ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ #3160

Complements

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

N	H ≅ G/N	type
3. #11729139	C₂	semidirect
4. #11729138	C₂	semidirect
5. #11729109	C₂	semidirect
62. #6386	Q₈	semidirect
102. C₇⋊C₁₆	C₂×Q₈	semidirect
254. C₇	#33731	semidirect

Maximal subgroups

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

subgroup	index
C₇×#3160	2
C₂×#89454	2
#11729139	2
#11729138	2
#11729109	2
#11728403	2
C₄×#6386	2
#33731	7