Subgroups — #13509359

Order 1792.

558 conjugacy classes of subgroups.

The subgroup lattice has 332 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, complemented
2	#11735915	C₂	1	1	yes	maximal, characteristic
3	#11729139	C₂	1	1	yes	maximal, characteristic, complemented
4	#11729135	C₂	1	1	yes	maximal, characteristic, complemented
5	#11729067	C₂	1	1	yes	maximal, characteristic, complemented
6	#11729064	C₂	1	1	yes	maximal, characteristic, complemented
7	#11728836	C₂	1	4	yes	maximal, complemented
8	C₄×#6402	C₂	1	4	yes	maximal, complemented
9	C₂×C₄×C₇⋊C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
10	C₇×#533	—	2	1	no	—
11	#90168	—	2	1	no	—
12	C₇×#500	C₂²	1	1	yes	characteristic
13	C₇×#494	C₂²	1	1	yes	characteristic
14	C₇×#487	—	2	1	no	—
15	C₇×#467	—	2	1	no	—
16	C₇×#467	C₂²	1	2	yes	—
17	C₄×C₇×Q₁₆	—	2	1	no	—
18	C₄×C₇×Q₁₆	C₂²	1	2	yes	—
19	C₂×C₄×C₇×C₈	C₂²	1	1	yes	characteristic
20	C₂×#6402	C₂²	1	4	yes	—
21	#89887	—	2	1	no	—
22	#89887	C₂²	1	2	yes	—
23	C₂²×C₇⋊C₁₆	C₂²	1	2	yes	—
24	#89467	C₂²	1	1	yes	characteristic
25	#89466	—	2	1	no	—
26	C₄×D₅₆	—	2	1	no	—
27	C₄×D₅₆	C₂²	1	2	yes	—
28	#89454	C₂²	1	1	yes	characteristic
29	#89359	C₂²	1	4	yes	—
30	#89286	C₂²	1	4	yes	—
31	#89285	C₂²	1	4	yes	—
32	C₄×C₇⋊C₁₆	C₂²	1	2	yes	—
33	C₄×C₇⋊C₁₆	C₂²	1	1	yes	characteristic
34	C₄×C₇⋊C₁₆	C₂²	1	1	yes	characteristic
35	#33736	—	7	1	no	maximal, Sylow
36	C₂×#1157	—	4	1	no	—
37	C₂×C₇×Q₁₆	—	2	1	no	—
38	C₂×C₇×Q₁₆	C₂³	1	2	yes	—
39	C₂²×C₇×C₈	D₄	1	1	yes	characteristic
40	C₂²×C₇×C₈	C₂³	1	1	yes	characteristic
41	C₇×C₄⋊Q₈	—	2	1	no	—
42	C₇×C₄⋊Q₈	—	2	1	no	—
43	C₇×#125	—	4	1	no	—
44	C₇×C₂²⋊Q₈	—	2	2	no	—
45	C₇×C₂²⋊Q₈	—	4	1	no	—
46	C₄×C₇×Q₈	—	2	2	no	—
47	C₄×C₇×Q₈	—	4	1	no	—
48	C₇×#117	—	4	1	no	—
49	C₂×C₄²×C₇	D₄	1	1	yes	characteristic
50	#6493	—	2	2	no	—
51	#6493	—	4	1	no	—
52	C₂×D₅₆	—	2	1	no	—
53	C₂×D₅₆	C₂³	1	2	yes	—
54	#6438	—	4	1	no	—
55	#6437	—	2	1	no	—
56	C₄×D₂₈	—	2	2	no	—
57	C₄×D₂₈	—	4	1	no	—
58	#6432	—	2	1	no	—
59	C₇×#107	—	2	1	no	—
60	C₇×#107	C₂³	1	2	yes	—
61	C₇×#103	—	2	2	no	—
62	C₇×#103	—	2	2	no	—
63	C₄×C₇×C₈	D₄	1	1	yes	characteristic
64	C₄×C₇×C₈	D₄	1	1	yes	characteristic
65	C₄×C₇×C₈	C₂³	1	1	yes	characteristic
66	C₄×C₇×C₈	C₂³	1	1	yes	characteristic
67	#6402	—	2	8	no	—
68	#6395	—	2	2	no	—
69	#6395	—	2	2	no	—
70	#6392	—	2	1	no	—
71	#6392	C₂³	1	2	yes	—
72	C₂×C₇⋊C₁₆	—	2	4	no	—
73	C₂×C₇⋊C₁₆	C₂³	1	2	yes	—
74	C₂×C₇⋊C₁₆	C₂³	1	2	yes	—
75	#3170	D₇	1	1	yes	characteristic, complemented
76	#3169	—	7	1	no	—
77	#2254	—	7	1	no	—
78	#2248	—	7	1	no	—
79	#2241	—	7	1	no	—
80	#2240	—	7	1	no	—
81	#2217	—	7	4	no	—
82	C₄×SD₃₂	—	7	4	no	—
83	C₂×C₄×C₁₆	—	7	1	no	—
84	C₂×C₇×Q₈	—	2	2	no	—
85	C₂×C₇×Q₈	—	4	1	no	—
86	C₂²×C₄×C₇	D₈	1	1	yes	characteristic
87	C₂²×C₄×C₇	D₈	1	1	yes	characteristic
88	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
89	C₂×C₂²⋊D₇	—	4	2	no	—
90	#1157	—	8	4	no	—
91	C₂×D₂₈	—	2	2	no	—
92	C₂×D₂₈	—	4	1	no	—
93	C₂×D₂₈	—	4	1	no	—
94	C₂×C₄×D₇	—	4	2	no	—
95	C₂×Dic₁₄	—	4	1	no	—
96	C₇×Q₁₆	—	2	4	no	—
97	C₇×Q₁₆	—	4	2	no	—
98	C₂×C₇×C₈	—	2	2	no	—
99	C₂×C₇×C₈	—	2	2	no	—
100	C₂×C₇×C₈	C₂×D₄	1	1	yes	characteristic
101	C₂×C₇×C₈	C₂×D₄	1	1	yes	characteristic
102	C₂×C₇×C₈	C₂×D₄	1	1	yes	characteristic
103	C₂×C₇×C₈	C₂⁴	1	1	yes	characteristic
104	C₇×C₄⋊C₄	—	2	2	no	—
105	C₇×C₄⋊C₄	—	4	1	no	—
106	C₇×C₄⋊C₄	—	4	1	no	—
107	C₇×C₄⋊C₄	—	4	2	no	—
108	C₇×C₄⋊C₄	—	4	2	no	—
109	C₇×C₄⋊C₄	—	4	1	no	—
110	C₇×C₂²⋊C₄	—	4	2	no	—
111	C₄²×C₇	—	4	2	no	—
112	C₄²×C₇	D₈	1	1	yes	characteristic
113	C₄²×C₇	D₈	1	1	yes	characteristic
114	C₄²×C₇	C₂×D₄	1	1	yes	characteristic
115	C₄²×C₇	C₂×D₄	1	1	yes	characteristic
116	(C₂×D₇)⋊C₄	—	4	2	no	—
117	(C₂×D₇)⋊C₄	—	4	2	no	—
118	C₄⋊Dic₇	—	2	2	no	—
119	C₄⋊Dic₇	—	4	1	no	—
120	D₅₆	—	2	4	no	—
121	D₅₆	—	4	2	no	—
122	C₇⋊C₁₆	—	2	2	no	—
123	C₇⋊C₁₆	#41	1	4	yes	complemented
124	#533	—	2	1	no	—
125	#532	—	14	1	no	—
126	C₂×SD₃₂	—	7	4	no	—
127	C₂²×C₁₆	—	7	2	no	—
128	#500	—	7	1	no	—
129	#500	C₂×D₇	1	1	yes	characteristic
130	#494	C₂×D₇	1	1	yes	characteristic
131	#493	—	7	1	no	—
132	#487	—	2	1	no	—
133	#486	—	14	1	no	—
134	#467	—	2	1	no	—
135	#467	C₂×D₇	1	2	yes	—
136	#466	—	7	2	no	—
137	#466	—	14	1	no	—
138	C₄×Q₁₆	—	2	1	no	—
139	C₄×Q₁₆	C₂×D₇	1	2	yes	—
140	C₄×D₈	—	7	2	no	—
141	C₄×D₈	—	14	1	no	—
142	C₂×C₄×C₈	C₂×D₇	1	1	yes	characteristic
143	#367	—	7	4	no	—
144	#358	—	7	4	no	—
145	#357	—	7	4	no	—
146	C₄×C₁₆	—	7	2	no	—
147	C₄×C₁₆	—	7	1	no	—
148	C₄×C₁₆	—	7	1	no	—
149	C₂³×C₇	C₂×D₈	1	1	yes	characteristic
150	C₂²×D₇	—	4	2	no	—
151	C₇×Q₈	—	4	2	no	—
152	C₇×Q₈	—	4	2	no	—
153	C₇×Q₈	—	8	2	no	—
154	C₂×C₄×C₇	—	2	1	no	—
155	C₂×C₄×C₇	—	2	1	no	—
156	C₂×C₄×C₇	—	2	2	no	—
157	C₂×C₄×C₇	—	4	2	no	—
158	C₂×C₄×C₇	—	4	2	no	—
159	C₂×C₄×C₇	SD₃₂	1	4	yes	complemented
160	C₂×C₄×C₇	C₂×D₈	1	1	yes	characteristic
161	C₂×C₄×C₇	C₂×D₈	1	1	yes	characteristic
162	C₂×C₄×C₇	C₂×D₈	1	1	yes	characteristic
163	C₂×C₄×C₇	C₂×D₈	1	1	yes	characteristic
164	C₂×C₄×C₇	C₂×D₈	1	1	yes	characteristic
165	C₂×C₄×C₇	C₂²×D₄	1	1	yes	characteristic
166	C₂²⋊D₇	—	8	4	no	—
167	C₂×Dic₇	—	4	2	no	—
168	D₂₈	—	4	2	no	—
169	D₂₈	—	4	2	no	—
170	D₂₈	—	8	2	no	—
171	D₂₈	—	8	2	no	—
172	C₄×D₇	—	8	4	no	—
173	Dic₁₄	—	8	2	no	—
174	C₇×C₈	—	2	1	no	—
175	C₇×C₈	—	2	1	no	—
176	C₇×C₈	#135	1	2	yes	—
177	C₇×C₈	C₂×#41	1	2	yes	—
178	C₂×#41	—	28	1	no	—
179	C₂×Q₁₆	—	2	1	no	—
180	C₂×Q₁₆	C₂²×D₇	1	2	yes	—
181	C₂×D₈	—	7	2	no	—
182	C₂×D₈	—	14	1	no	—
183	C₂²×C₈	C₂²⋊D₇	1	1	yes	characteristic
184	C₂²×C₈	C₂²×D₇	1	1	yes	characteristic
185	C₄⋊Q₈	—	2	1	no	—
186	C₄⋊Q₈	—	2	1	no	—
187	C₄⋊Q₈	—	14	1	no	—
188	#127	—	14	1	no	—
189	#125	—	4	1	no	—
190	#124	—	28	1	no	—
191	C₂²⋊Q₈	—	2	2	no	—
192	C₂²⋊Q₈	—	4	1	no	—
193	#121	—	14	2	no	—
194	#121	—	28	1	no	—
195	C₄×Q₈	—	2	2	no	—
196	C₄×Q₈	—	4	1	no	—
197	C₄×D₄	—	14	2	no	—
198	C₄×D₄	—	28	1	no	—
199	#117	—	4	1	no	—
200	C₂×C₄²	C₂²⋊D₇	1	1	yes	characteristic
201	SD₃₂	—	14	8	no	—
202	C₂×C₁₆	—	7	2	no	—
203	C₂×C₁₆	—	7	2	no	—
204	C₂×C₁₆	—	14	4	no	—
205	#107	—	2	1	no	—
206	#107	—	7	2	no	—
207	#107	—	14	1	no	—
208	#107	C₂²×D₇	1	2	yes	—
209	#103	—	2	2	no	—
210	#103	—	2	2	no	—
211	#102	—	14	2	no	—
212	#102	—	14	2	no	—
213	C₄×C₈	C₂²⋊D₇	1	1	yes	characteristic
214	C₄×C₈	C₂²⋊D₇	1	1	yes	characteristic
215	C₄×C₈	C₂²×D₇	1	1	yes	characteristic
216	C₄×C₈	C₂²×D₇	1	1	yes	characteristic
217	C₂²×C₇	—	2	1	no	—
218	C₂²×C₇	—	2	1	no	—
219	C₂²×C₇	C₂×SD₃₂	1	2	yes	—
220	C₂²×C₇	C₂²×D₈	1	1	yes	characteristic
221	C₂×D₇	—	8	2	no	—
222	C₂×D₇	—	8	2	no	—
223	C₂×D₇	—	8	2	no	—
224	C₄×C₇	—	2	1	no	—
225	C₄×C₇	—	2	1	no	—
226	C₄×C₇	—	8	2	no	—
227	C₄×C₇	—	8	2	no	—
228	C₄×C₇	C₂×SD₃₂	1	2	yes	—
229	C₄×C₇	C₂×SD₃₂	1	2	yes	—
230	C₄×C₇	#508	1	2	yes	—
231	C₄×C₇	#532	1	1	yes	characteristic
232	C₄×C₇	C₂×#135	1	1	yes	characteristic
233	Dic₇	—	8	2	no	—
234	#41	—	56	4	no	—
235	C₂×Q₈	—	2	2	no	—
236	C₂×Q₈	—	4	1	no	—
237	C₂×Q₈	—	28	1	no	—
238	C₂×D₄	—	14	2	no	—
239	C₂×D₄	—	28	2	no	—
240	C₂×D₄	—	28	1	no	—
241	C₂×D₄	—	28	1	no	—
242	C₂²×C₄	—	28	2	no	—
243	C₂²×C₄	#1141	1	1	yes	characteristic
244	C₂²×C₄	#1141	1	1	yes	characteristic
245	C₂²×C₄	C₂×C₂²⋊D₇	1	1	yes	characteristic
246	Q₁₆	—	2	4	no	—
247	Q₁₆	—	4	2	no	—
248	D₈	—	14	4	no	—
249	D₈	—	28	2	no	—
250	C₂×C₈	—	2	2	no	—
251	C₂×C₈	—	2	2	no	—
252	C₂×C₈	C₂×C₂²⋊D₇	1	1	yes	characteristic
253	C₂×C₈	C₂×C₂²⋊D₇	1	1	yes	characteristic
254	C₂×C₈	C₂×C₂²⋊D₇	1	1	yes	characteristic
255	C₂×C₈	C₂³×D₇	1	1	yes	characteristic
256	C₄⋊C₄	—	2	2	no	—
257	C₄⋊C₄	—	4	1	no	—
258	C₄⋊C₄	—	4	1	no	—
259	C₄⋊C₄	—	4	2	no	—
260	C₄⋊C₄	—	4	2	no	—
261	C₄⋊C₄	—	4	1	no	—
262	C₄⋊C₄	—	14	2	no	—
263	C₄⋊C₄	—	28	1	no	—
264	C₂²⋊C₄	—	4	2	no	—
265	C₂²⋊C₄	—	28	2	no	—
266	C₂²⋊C₄	—	28	2	no	—
267	C₄²	—	4	2	no	—
268	C₄²	#1141	1	1	yes	characteristic
269	C₄²	#1141	1	1	yes	characteristic
270	C₄²	C₂×C₂²⋊D₇	1	1	yes	characteristic
271	C₄²	C₂×C₂²⋊D₇	1	1	yes	characteristic
272	C₁₆	—	7	4	no	—
273	C₁₆	—	14	2	no	—
274	C₂×C₇	—	2	2	no	—
275	C₂×C₇	#3169	1	1	yes	characteristic
276	C₂×C₇	C₂²×SD₃₂	1	1	yes	characteristic
277	C₂×C₇	C₂×#508	1	1	yes	characteristic
278	D₇	—	16	2	no	—
279	C₂³	—	28	2	no	—
280	C₂³	C₂×#1141	1	1	yes	characteristic
281	Q₈	—	4	2	no	—
282	Q₈	—	4	2	no	—
283	Q₈	—	8	2	no	—
284	Q₈	—	56	2	no	—
285	D₄	—	28	2	no	—
286	D₄	—	28	2	no	—
287	D₄	—	56	4	no	—
288	D₄	—	56	2	no	—
289	D₄	—	56	2	no	—
290	C₂×C₄	—	2	2	no	—
291	C₂×C₄	—	2	1	no	—
292	C₂×C₄	—	2	1	no	—
293	C₂×C₄	—	4	2	no	—
294	C₂×C₄	—	4	2	no	—
295	C₂×C₄	—	28	2	no	—
296	C₂×C₄	—	56	4	no	—
297	C₂×C₄	#6402	1	4	yes	complemented
298	C₂×C₄	C₂×#1141	1	1	yes	characteristic
299	C₂×C₄	C₂×#1141	1	1	yes	characteristic
300	C₂×C₄	C₂×#1141	1	1	yes	characteristic
301	C₂×C₄	C₂×#1141	1	1	yes	characteristic
302	C₂×C₄	C₂×#1141	1	1	yes	characteristic
303	C₂×C₄	C₂²×C₂²⋊D₇	1	1	yes	characteristic
304	C₈	—	2	1	no	—
305	C₈	—	2	1	no	—
306	C₈	#6513	1	2	yes	—
307	C₈	D₇×#41	1	2	yes	—
308	C₇	#33736	1	1	yes	characteristic, complemented, Sylow
309	C₂²	—	2	1	no	—
310	C₂²	—	2	1	no	—
311	C₂²	—	56	2	no	—
312	C₂²	—	56	2	no	—
313	C₂²	—	56	2	no	—
314	C₂²	C₂×#6402	1	2	yes	—
315	C₂²	C₂²×#1141	1	1	yes	characteristic
316	C₄	—	2	1	no	—
317	C₄	—	2	1	no	—
318	C₄	—	8	2	no	—
319	C₄	—	8	2	no	—
320	C₄	—	56	2	no	—
321	C₄	C₂×#6402	1	2	yes	—
322	C₄	C₂×#6402	1	2	yes	—
323	C₄	#89966	1	2	yes	—
324	C₄	C₂×#6513	1	1	yes	characteristic
325	C₄	#90523	1	1	yes	characteristic
326	C₂	—	2	2	no	—
327	C₂	—	112	2	no	—
328	C₂	#11740736	1	1	yes	characteristic
329	C₂	C₂²×#6402	1	1	yes	characteristic
330	C₂	C₂×#89966	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×#1141
commutator, cosocle	C₂×C₇×C₈	C₂⁴
Frattini	C₂×C₈	C₂³×D₇
Fitting	C₇×#3170	C₂
socle	C₂²×C₇	C₂²×D₈
prime radical	#11735915	C₂
hypercenter	#3170	D₇
hypocenter	C₇	#33736
LCS	G ▷ C₂×C₇×C₈ ▷ C₄×C₇ ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ #3170

Complements

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

N	H ≅ G/N	type
1. C₇×#3170	C₂	semidirect
3. #11729139	C₂	semidirect
4. #11729135	C₂	semidirect
5. #11729067	C₂	semidirect
6. #11729064	C₂	semidirect
7. #11728836	C₂	semidirect
8. C₄×#6402	C₂	semidirect
9. C₂×C₄×C₇⋊C₁₆	C₂	semidirect
75. #3170	D₇	semidirect
123. C₇⋊C₁₆	#41	semidirect
159. C₂×C₄×C₇	SD₃₂	semidirect
297. C₂×C₄	#6402	semidirect
308. C₇	#33736	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
#11735915	2
#11729139	2
#11729135	2
#11729067	2
#11729064	2
#11728836	2
C₄×#6402	2
C₂×C₄×C₇⋊C₁₆	2
#33736	7