Subgroups — #13509315

Order 1792.

514 conjugacy classes of subgroups.

The subgroup lattice has 308 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#11739383	C₂	1	1	yes	maximal, characteristic
2	D₇×#500	C₂	1	1	yes	maximal, characteristic
3	C₇×#2252	C₂	1	1	yes	maximal, characteristic, complemented
4	#11729139	C₂	1	1	yes	maximal, characteristic, complemented
5	#11729138	C₂	1	1	yes	maximal, characteristic, complemented
6	#11727080	C₂	1	4	yes	maximal
7	#11727076	C₂	1	4	yes	maximal, complemented
8	C₄×#6372	C₂	1	1	yes	maximal, characteristic
9	#11726800	C₂	1	1	yes	maximal, characteristic, complemented
10	#90419	—	2	1	no	—
11	D₇×C₄⋊Q₈	—	2	1	no	—
12	C₇×#500	C₂²	1	1	yes	characteristic
13	C₇×#500	C₂²	1	1	yes	characteristic
14	#89851	—	2	1	no	—
15	#89850	—	2	1	no	—
16	C₂×#6372	C₂²	1	2	yes	—
17	#89652	—	2	1	no	—
18	#89652	C₂²	1	2	yes	—
19	D₇×#107	—	2	1	no	—
20	D₇×#107	C₂²	1	2	yes	—
21	#89650	—	2	1	no	—
22	#89650	C₂²	1	2	yes	—
23	#89646	—	2	1	no	—
24	#89646	C₂²	1	2	yes	—
25	C₄×C₈×D₇	C₂²	1	1	yes	characteristic
26	#89454	C₂²	1	1	yes	characteristic
27	#89454	C₂²	1	1	yes	characteristic
28	C₇×#366	C₂²	1	4	yes	—
29	C₄×C₇×C₁₆	C₂²	1	1	yes	characteristic
30	#89300	C₂²	1	4	yes	—
31	#89297	C₂²	1	2	yes	—
32	#89285	C₂²	1	4	yes	—
33	#89284	C₂²	1	4	yes	—
34	C₄×C₇⋊C₁₆	C₂²	1	1	yes	characteristic
35	#33731	—	7	1	no	maximal, Sylow
36	C₂×Q₈×D₇	—	4	1	no	—
37	C₇×C₄⋊Q₈	—	2	1	no	—
38	C₇×C₄⋊Q₈	—	2	1	no	—
39	#6509	—	4	1	no	—
40	Q₈×Dic₇	—	4	1	no	—
41	#6507	—	4	1	no	—
42	C₂×C₈×D₇	D₄	1	1	yes	characteristic
43	C₂×C₈×D₇	C₂³	1	1	yes	characteristic
44	#6460	—	4	1	no	—
45	#6455	—	2	2	no	—
46	D₇×C₄⋊C₄	—	2	2	no	—
47	D₇×C₄⋊C₄	—	4	1	no	—
48	#6453	—	2	2	no	—
49	#6451	—	2	2	no	—
50	#6451	—	4	1	no	—
51	#6450	—	4	1	no	—
52	C₄²×D₇	D₄	1	1	yes	characteristic
53	#6432	—	2	1	no	—
54	#6432	—	2	1	no	—
55	C₂×C₇×C₁₆	C₂³	1	2	yes	—
56	C₇×#107	—	2	1	no	—
57	C₇×#107	—	2	1	no	—
58	C₇×#107	C₂³	1	2	yes	—
59	C₇×#107	C₂³	1	2	yes	—
60	C₄×C₇×C₈	C₂³	1	1	yes	characteristic
61	#6392	—	2	1	no	—
62	#6392	—	2	1	no	—
63	#6392	C₂³	1	2	yes	—
64	#6392	C₂³	1	2	yes	—
65	C₈×Dic₇	D₄	1	1	yes	characteristic
66	C₈×Dic₇	C₂³	1	1	yes	characteristic
67	C₂×C₇⋊C₁₆	C₂³	1	2	yes	—
68	#6382	—	2	2	no	—
69	#6382	—	2	2	no	—
70	#6381	—	2	2	no	—
71	#6381	—	2	2	no	—
72	C₄×C₇⋊C₈	D₄	1	1	yes	characteristic
73	#6372	Q₈	1	8	yes	complemented
74	#3160	—	7	1	no	—
75	C₂×#500	—	7	1	no	—
76	#2255	—	7	4	no	—
77	#2254	—	7	1	no	—
78	#2254	—	7	1	no	—
79	#2252	—	7	1	no	—
80	#2252	D₇	1	1	yes	characteristic, complemented
81	#2159	—	7	4	no	—
82	C₄×M₃₂	—	7	1	no	—
83	C₂×C₇×Q₈	—	4	1	no	—
84	C₂×C₇×Q₈	—	4	1	no	—
85	Q₈×D₇	—	8	4	no	—
86	C₂×C₄×D₇	—	4	2	no	—
87	C₂×C₄×D₇	D₈	1	1	yes	characteristic
88	C₂×C₄×D₇	D₈	1	1	yes	characteristic
89	C₂×C₄×D₇	C₂×D₄	1	1	yes	characteristic
90	C₂×Dic₁₄	—	4	1	no	—
91	C₂×Dic₁₄	—	4	1	no	—
92	C₂×Dic₁₄	—	4	2	no	—
93	C₂×C₇×C₈	C₂×D₄	1	1	yes	characteristic
94	C₂×C₇×C₈	C₂⁴	1	1	yes	characteristic
95	C₇×C₄⋊C₄	—	2	2	no	—
96	C₇×C₄⋊C₄	—	2	2	no	—
97	C₇×C₄⋊C₄	—	4	1	no	—
98	C₇×C₄⋊C₄	—	4	1	no	—
99	C₄²×C₇	C₂×D₄	1	1	yes	characteristic
100	(C₂×D₇)⋊C₄	—	4	2	no	—
101	C₄⋊Dic₇	—	2	2	no	—
102	C₄⋊Dic₇	—	2	2	no	—
103	C₄⋊Dic₇	—	4	1	no	—
104	C₄⋊Dic₇	—	4	1	no	—
105	C₄⋊Dic₇	—	4	2	no	—
106	Dic₇⋊C₄	—	4	2	no	—
107	Dic₇⋊C₄	—	4	2	no	—
108	Dic₇⋊C₄	—	4	2	no	—
109	C₄×Dic₇	—	4	2	no	—
110	C₄×Dic₇	D₈	1	1	yes	characteristic
111	C₄×Dic₇	D₈	1	1	yes	characteristic
112	C₄×Dic₇	C₂×D₄	1	1	yes	characteristic
113	C₂×C₇⋊C₈	C₂×D₄	1	1	yes	characteristic
114	C₂×C₇⋊C₈	C₂×D₄	1	1	yes	characteristic
115	C₈×D₇	Q₁₆	1	4	yes	—
116	C₈×D₇	C₂×Q₈	1	4	yes	—
117	C₇×C₁₆	C₂×Q₈	1	4	yes	complemented
118	C₇⋊C₁₆	C₂×Q₈	1	4	yes	complemented
119	#533	—	14	1	no	—
120	C₂×C₄⋊Q₈	—	14	1	no	—
121	C₂×M₃₂	—	7	2	no	—
122	#500	—	7	2	no	—
123	#500	—	7	1	no	—
124	#500	—	7	1	no	—
125	#500	—	14	1	no	—
126	#500	—	14	1	no	—
127	#500	C₂×D₇	1	1	yes	characteristic
128	#500	C₂×D₇	1	1	yes	characteristic
129	#499	—	7	2	no	—
130	#499	—	14	1	no	—
131	#498	—	14	1	no	—
132	#427	—	7	2	no	—
133	#427	—	14	1	no	—
134	C₂×#107	—	7	2	no	—
135	C₂×#107	—	14	1	no	—
136	C₂×C₄×C₈	—	7	1	no	—
137	#367	—	7	4	no	—
138	#367	—	7	4	no	—
139	#366	—	7	4	no	—
140	#366	C₂×D₇	1	4	yes	—
141	#346	—	7	2	no	—
142	C₄×C₁₆	—	7	1	no	—
143	C₄×C₁₆	C₂×D₇	1	1	yes	characteristic
144	C₂²×D₇	C₂×D₈	1	1	yes	characteristic
145	C₇×Q₈	—	8	2	no	—
146	C₇×Q₈	—	8	2	no	—
147	C₂×C₄×C₇	—	4	2	no	—
148	C₂×C₄×C₇	—	4	2	no	—
149	C₂×C₄×C₇	C₂×D₈	1	1	yes	characteristic
150	C₂×C₄×C₇	C₂×D₈	1	1	yes	characteristic
151	C₂×C₄×C₇	C₂²×D₄	1	1	yes	characteristic
152	C₂×Dic₇	—	4	2	no	—
153	C₂×Dic₇	—	4	2	no	—
154	C₂×Dic₇	C₂×D₈	1	1	yes	characteristic
155	C₂×Dic₇	C₂×D₈	1	1	yes	characteristic
156	C₂×Dic₇	C₂×D₈	1	1	yes	characteristic
157	C₄×D₇	—	2	2	no	—
158	C₄×D₇	—	2	2	no	—
159	C₄×D₇	—	8	4	no	—
160	C₄×D₇	C₄⋊Q₈	1	2	yes	—
161	C₄×D₇	C₂×Q₁₆	1	2	yes	—
162	Dic₁₄	—	8	4	no	—
163	Dic₁₄	—	8	2	no	—
164	Dic₁₄	—	8	2	no	—
165	C₇×C₈	C₂×Q₁₆	1	2	yes	—
166	C₇×C₈	C₂²×Q₈	1	2	yes	—
167	C₇⋊C₈	C₄⋊Q₈	1	2	yes	—
168	C₇⋊C₈	C₂×Q₁₆	1	2	yes	—
169	C₂²×Q₈	—	28	1	no	—
170	C₂²×C₈	—	7	1	no	—
171	C₂²×C₈	—	7	1	no	—
172	C₄⋊Q₈	—	2	1	no	—
173	C₄⋊Q₈	—	2	1	no	—
174	C₄⋊Q₈	—	14	2	no	—
175	C₄⋊Q₈	—	14	1	no	—
176	C₄⋊Q₈	—	14	1	no	—
177	C₄⋊Q₈	—	28	1	no	—
178	C₄⋊Q₈	—	28	1	no	—
179	#125	—	14	2	no	—
180	C₂²⋊Q₈	—	28	1	no	—
181	C₂²⋊Q₈	—	28	1	no	—
182	C₄×Q₈	—	28	1	no	—
183	C₄×Q₈	—	28	1	no	—
184	#117	—	14	2	no	—
185	C₂×C₄⋊C₄	—	14	2	no	—
186	C₂×C₄⋊C₄	—	28	1	no	—
187	C₂×C₄²	—	7	1	no	—
188	M₃₂	—	7	8	no	—
189	C₂×C₁₆	—	7	2	no	—
190	C₂×C₁₆	C₂²×D₇	1	2	yes	—
191	#107	—	2	1	no	—
192	#107	—	2	1	no	—
193	#107	—	7	2	no	—
194	#107	—	7	2	no	—
195	#107	—	14	2	no	—
196	#107	—	14	2	no	—
197	#107	—	14	1	no	—
198	#107	—	14	1	no	—
199	#107	C₂²×D₇	1	2	yes	—
200	#107	C₂²×D₇	1	2	yes	—
201	#106	—	14	2	no	—
202	#106	—	14	2	no	—
203	C₄×C₈	—	7	1	no	—
204	C₄×C₈	—	7	1	no	—
205	C₄×C₈	—	7	1	no	—
206	C₄×C₈	C₂²×D₇	1	1	yes	characteristic
207	C₂²×C₇	C₂²×D₈	1	1	yes	characteristic
208	C₂×D₇	—	2	1	no	—
209	C₂×D₇	—	2	1	no	—
210	C₂×D₇	#500	1	2	yes	—
211	C₄×C₇	—	8	2	no	—
212	C₄×C₇	—	8	2	no	—
213	C₄×C₇	#509	1	2	yes	—
214	C₄×C₇	#510	1	2	yes	—
215	C₄×C₇	C₂×C₄⋊Q₈	1	1	yes	characteristic
216	C₄×C₇	C₂²×Q₁₆	1	1	yes	characteristic
217	Dic₇	—	2	1	no	—
218	Dic₇	—	2	1	no	—
219	Dic₇	—	8	2	no	—
220	Dic₇	—	8	2	no	—
221	Dic₇	#500	1	2	yes	—
222	C₂×Q₈	—	4	1	no	—
223	C₂×Q₈	—	4	1	no	—
224	C₂×Q₈	—	28	2	no	—
225	C₂×Q₈	—	28	1	no	—
226	C₂×Q₈	—	28	1	no	—
227	C₂×Q₈	—	56	4	no	—
228	C₂²×C₄	—	7	1	no	—
229	C₂²×C₄	—	7	1	no	—
230	C₂²×C₄	—	7	1	no	—
231	C₂²×C₄	—	28	2	no	—
232	C₂×C₈	—	7	1	no	—
233	C₂×C₈	—	7	1	no	—
234	C₂×C₈	—	7	4	no	—
235	C₂×C₈	—	7	4	no	—
236	C₂×C₈	D₄×D₇	1	1	yes	characteristic
237	C₂×C₈	C₂³×D₇	1	1	yes	characteristic
238	C₄⋊C₄	—	2	2	no	—
239	C₄⋊C₄	—	2	2	no	—
240	C₄⋊C₄	—	4	1	no	—
241	C₄⋊C₄	—	4	1	no	—
242	C₄⋊C₄	—	14	2	no	—
243	C₄⋊C₄	—	14	2	no	—
244	C₄⋊C₄	—	28	2	no	—
245	C₄⋊C₄	—	28	2	no	—
246	C₄⋊C₄	—	28	2	no	—
247	C₄⋊C₄	—	28	2	no	—
248	C₄⋊C₄	—	28	1	no	—
249	C₄⋊C₄	—	28	1	no	—
250	C₂²⋊C₄	—	28	2	no	—
251	C₄²	—	7	1	no	—
252	C₄²	—	7	1	no	—
253	C₄²	—	7	1	no	—
254	C₄²	—	28	2	no	—
255	C₄²	D₄×D₇	1	1	yes	characteristic
256	C₁₆	—	7	4	no	—
257	C₁₆	Q₈×D₇	1	4	yes	complemented
258	C₂×C₇	C₂×#500	1	1	yes	characteristic
259	C₂×C₇	C₂×#509	1	1	yes	characteristic
260	C₂×C₇	C₂×#510	1	1	yes	characteristic
261	D₇	—	2	2	no	—
262	C₂³	—	7	1	no	—
263	Q₈	—	8	2	no	—
264	Q₈	—	8	2	no	—
265	Q₈	—	56	4	no	—
266	Q₈	—	56	2	no	—
267	Q₈	—	56	2	no	—
268	C₂×C₄	—	4	2	no	—
269	C₂×C₄	—	4	2	no	—
270	C₂×C₄	—	7	1	no	—
271	C₂×C₄	—	7	2	no	—
272	C₂×C₄	—	7	2	no	—
273	C₂×C₄	—	7	1	no	—
274	C₂×C₄	—	7	1	no	—
275	C₂×C₄	—	14	2	no	—
276	C₂×C₄	—	14	2	no	—
277	C₂×C₄	—	28	2	no	—
278	C₂×C₄	—	28	2	no	—
279	C₂×C₄	—	56	4	no	—
280	C₂×C₄	D₇×D₈	1	1	yes	characteristic
281	C₂×C₄	D₇×D₈	1	1	yes	characteristic
282	C₂×C₄	C₂×D₄×D₇	1	1	yes	characteristic
283	C₈	—	7	2	no	—
284	C₈	—	7	2	no	—
285	C₈	D₇×Q₁₆	1	2	yes	—
286	C₈	C₂×Q₈×D₇	1	2	yes	—
287	C₇	#33731	1	1	yes	characteristic, complemented, Sylow
288	C₂²	—	7	2	no	—
289	C₂²	—	14	1	no	—
290	C₂²	—	14	1	no	—
291	C₂²	C₂×D₇×D₈	1	1	yes	characteristic
292	C₄	—	7	2	no	—
293	C₄	—	8	2	no	—
294	C₄	—	8	2	no	—
295	C₄	—	14	1	no	—
296	C₄	—	14	1	no	—
297	C₄	—	56	2	no	—
298	C₄	—	56	2	no	—
299	C₄	#89683	1	2	yes	—
300	C₄	#89690	1	2	yes	—
301	C₄	D₇×C₄⋊Q₈	1	1	yes	characteristic
302	C₄	C₂×D₇×Q₁₆	1	1	yes	characteristic
303	C₂	—	14	2	no	—
304	C₂	D₇×#500	1	1	yes	characteristic
305	C₂	C₂×#89683	1	1	yes	characteristic
306	C₂	C₂×#89690	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₇×D₈
commutator, cosocle	C₂×C₇×C₈	C₂⁴
Frattini	C₂×C₈	C₂³×D₇
Fitting	C₇×#2252	C₂
socle	C₂²×C₇	C₂²×D₈
prime radical	C₂²×D₇	C₂×D₈
hypercenter	#2252	D₇
hypocenter	C₇	#33731
LCS	G ▷ C₂×C₇×C₈ ▷ C₄×C₇ ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #2252

Complements

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

N	H ≅ G/N	type
3. C₇×#2252	C₂	semidirect
4. #11729139	C₂	semidirect
5. #11729138	C₂	semidirect
7. #11727076	C₂	semidirect
9. #11726800	C₂	semidirect
73. #6372	Q₈	semidirect
80. #2252	D₇	semidirect
117. C₇×C₁₆	C₂×Q₈	semidirect
118. C₇⋊C₁₆	C₂×Q₈	semidirect
257. C₁₆	Q₈×D₇	semidirect
287. C₇	#33731	semidirect

Maximal subgroups

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

subgroup	index
#11739383	2
D₇×#500	2
C₇×#2252	2
#11729139	2
#11729138	2
#11727080	2
#11727076	2
C₄×#6372	2
#11726800	2
#33731	7