Subgroups — #13273656

Order 1792.

342 conjugacy classes of subgroups.

The subgroup lattice has 320 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×(C₂×D₇)⋊C₁₆	C₂	1	1	yes	maximal, characteristic
2	#11728384	C₂	1	1	yes	maximal, characteristic
3	#11728384	C₂	1	1	yes	maximal, characteristic
4	C₂×#89284	C₂	1	1	yes	maximal, characteristic, complemented
5	C₇×#1380	C₂	1	1	yes	maximal, characteristic, complemented
6	#11725612	C₂	1	1	yes	maximal, characteristic, complemented
7	#11725528	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×C₈×D₇	C₂²	1	1	yes	characteristic
9	C₂²×C₇×C₁₆	C₂²	1	1	yes	characteristic
10	C₂×C₇×#107	C₂²	1	1	yes	characteristic
11	C₂×C₇×#107	C₂²	1	1	yes	characteristic
12	C₂×#6392	C₂²	1	1	yes	characteristic
13	C₂×#6392	C₂²	1	1	yes	characteristic
14	C₂²×C₇⋊C₁₆	C₂²	1	1	yes	characteristic
15	#89759	—	2	1	no	—
16	#89759	—	2	1	no	—
17	#89324	—	2	1	no	—
18	#89324	—	2	1	no	—
19	(C₂×D₇)⋊C₁₆	—	2	2	no	—
20	#89284	—	2	2	no	—
21	#13492	—	7	1	no	maximal, Sylow
22	C₂²×C₄×D₇	D₄	1	1	yes	characteristic
23	C₂²×C₇×C₈	C₂³	1	1	yes	characteristic
24	C₂×C₇×C₄⋊C₄	—	2	1	no	—
25	C₂×C₇×C₄⋊C₄	—	2	1	no	—
26	C₂×(C₂×D₇)⋊C₄	—	4	1	no	—
27	C₂×(C₂×D₇)⋊C₄	—	4	1	no	—
28	C₂×C₄⋊Dic₇	—	2	1	no	—
29	C₂×C₄⋊Dic₇	—	2	1	no	—
30	C₂²×C₇⋊C₈	D₄	1	1	yes	characteristic
31	C₂×C₈×D₇	—	2	1	no	—
32	C₂×C₈×D₇	—	2	1	no	—
33	C₂×C₈×D₇	—	2	1	no	—
34	C₂×C₈×D₇	—	2	1	no	—
35	C₂×C₈×D₇	—	2	1	no	—
36	C₂×C₈×D₇	—	2	1	no	—
37	C₂×C₇×C₁₆	—	2	1	no	—
38	C₂×C₇×C₁₆	—	2	1	no	—
39	C₂×C₇×C₁₆	—	2	1	no	—
40	C₇×#107	—	2	1	no	—
41	C₇×#107	—	2	1	no	—
42	C₇×#107	—	2	1	no	—
43	C₇×#107	—	2	1	no	—
44	C₇⋊#95	—	4	1	no	—
45	C₇⋊#95	—	4	1	no	—
46	#6392	—	2	1	no	—
47	#6392	—	2	1	no	—
48	#6392	—	2	1	no	—
49	#6392	—	2	1	no	—
50	C₂×C₇⋊C₁₆	—	2	1	no	—
51	C₂×C₇⋊C₁₆	—	2	1	no	—
52	C₂×C₇⋊C₁₆	D₄	1	2	yes	—
53	C₂×#366	—	7	1	no	—
54	C₂×C₂²⋊C₁₆	—	7	1	no	—
55	#1808	—	7	1	no	—
56	#1808	—	7	1	no	—
57	#1380	—	7	1	no	—
58	#1380	—	7	1	no	—
59	#1380	D₇	1	1	yes	characteristic, complemented
60	C₂³×D₇	D₈	1	1	yes	characteristic
61	C₂²×C₄×C₇	—	4	1	no	—
62	C₂²×C₄×C₇	—	4	1	no	—
63	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
64	C₂²×Dic₇	—	4	1	no	—
65	C₂²×Dic₇	—	4	1	no	—
66	C₂²×Dic₇	D₈	1	1	yes	characteristic
67	C₂×C₄×D₇	—	2	1	no	—
68	C₂×C₄×D₇	—	2	1	no	—
69	C₂×C₄×D₇	—	2	1	no	—
70	C₂×C₄×D₇	—	2	1	no	—
71	C₂×C₄×D₇	—	2	1	no	—
72	C₂×C₄×D₇	—	2	1	no	—
73	C₂×C₇×C₈	C₂×D₄	1	1	yes	characteristic
74	C₂×C₇×C₈	#41	1	1	yes	characteristic
75	C₂×C₇×C₈	#41	1	1	yes	characteristic
76	C₂×C₇×C₈	#41	1	1	yes	characteristic
77	C₂×C₇×C₈	#41	1	1	yes	characteristic
78	C₂×C₇×C₈	#41	1	1	yes	characteristic
79	C₇×C₄⋊C₄	—	4	1	no	—
80	C₇×C₄⋊C₄	—	4	1	no	—
81	C₇×C₄⋊C₄	—	4	1	no	—
82	C₇×C₄⋊C₄	—	4	1	no	—
83	(C₂×D₇)⋊C₄	—	8	2	no	—
84	(C₂×D₇)⋊C₄	—	8	2	no	—
85	C₄⋊Dic₇	—	4	1	no	—
86	C₄⋊Dic₇	—	4	1	no	—
87	C₄⋊Dic₇	—	4	1	no	—
88	C₄⋊Dic₇	—	4	1	no	—
89	C₂×C₇⋊C₈	—	2	1	no	—
90	C₂×C₇⋊C₈	—	2	1	no	—
91	C₂×C₇⋊C₈	—	2	1	no	—
92	C₈×D₇	—	4	1	no	—
93	C₈×D₇	—	4	1	no	—
94	C₈×D₇	—	4	1	no	—
95	C₈×D₇	—	4	1	no	—
96	C₇×C₁₆	—	4	1	no	—
97	C₇⋊C₁₆	—	2	2	no	—
98	C₂³×C₈	—	7	1	no	—
99	C₂²×C₁₆	—	7	1	no	—
100	C₂²×C₁₆	C₂×D₇	1	1	yes	characteristic
101	C₂×#107	—	7	1	no	—
102	C₂×#107	—	7	1	no	—
103	C₂×#107	C₂×D₇	1	1	yes	characteristic
104	C₂×#107	C₂×D₇	1	1	yes	characteristic
105	#380	—	14	1	no	—
106	#380	—	14	1	no	—
107	#366	—	14	2	no	—
108	C₂²⋊C₁₆	—	14	2	no	—
109	#340	—	14	1	no	—
110	#340	—	14	1	no	—
111	C₂³×C₇	C₂×D₈	1	1	yes	characteristic
112	C₂²×D₇	—	2	1	no	—
113	C₂²×D₇	—	2	1	no	—
114	C₂²×D₇	—	2	1	no	—
115	C₂²×D₇	—	2	1	no	—
116	C₂²×D₇	—	2	1	no	—
117	C₂²×D₇	—	2	1	no	—
118	C₂²×D₇	D₁₆	1	2	yes	—
119	C₂×C₄×C₇	—	8	1	no	—
120	C₂×C₄×C₇	—	8	1	no	—
121	C₂×C₄×C₇	—	8	1	no	—
122	C₂×C₄×C₇	—	8	1	no	—
123	C₂×C₄×C₇	—	8	1	no	—
124	C₂×C₄×C₇	—	8	1	no	—
125	C₂×C₄×C₇	#121	1	1	yes	characteristic
126	C₂×C₄×C₇	#123	1	1	yes	characteristic
127	C₂×C₄×C₇	#123	1	1	yes	characteristic
128	C₂×C₄×C₇	#135	1	1	yes	characteristic
129	C₂×C₄×C₇	#137	1	1	yes	characteristic
130	C₂×C₄×C₇	#137	1	1	yes	characteristic
131	C₂×Dic₇	—	2	1	no	—
132	C₂×Dic₇	—	2	1	no	—
133	C₂×Dic₇	—	2	1	no	—
134	C₂×Dic₇	—	8	1	no	—
135	C₂×Dic₇	—	8	1	no	—
136	C₂×Dic₇	—	8	1	no	—
137	C₂×Dic₇	—	8	1	no	—
138	C₂×Dic₇	—	8	1	no	—
139	C₂×Dic₇	—	8	1	no	—
140	C₄×D₇	—	4	1	no	—
141	C₄×D₇	—	4	1	no	—
142	C₄×D₇	—	4	1	no	—
143	C₄×D₇	—	4	1	no	—
144	C₇×C₈	#123	1	1	yes	characteristic
145	C₇×C₈	#124	1	1	yes	characteristic
146	C₇×C₈	#126	1	1	yes	characteristic
147	C₇×C₈	#126	1	1	yes	characteristic
148	C₇⋊C₈	—	4	1	no	—
149	C₂³×C₄	—	7	1	no	—
150	C₂²×C₈	—	7	1	no	—
151	C₂²×C₈	—	14	1	no	—
152	C₂²×C₈	—	14	1	no	—
153	C₂²×C₈	—	14	1	no	—
154	C₂²×C₈	—	14	1	no	—
155	C₂²×C₈	—	14	1	no	—
156	C₂²×C₈	—	14	1	no	—
157	C₂²×C₈	C₂²×D₇	1	1	yes	characteristic
158	C₂×C₄⋊C₄	—	2	1	no	—
159	C₂×C₄⋊C₄	—	2	1	no	—
160	C₂×C₄⋊C₄	—	14	1	no	—
161	C₂×C₄⋊C₄	—	14	1	no	—
162	C₂×C₂²⋊C₄	—	28	1	no	—
163	C₂×C₂²⋊C₄	—	28	1	no	—
164	C₂×C₁₆	—	2	1	no	—
165	C₂×C₁₆	—	2	1	no	—
166	C₂×C₁₆	—	2	1	no	—
167	C₂×C₁₆	—	7	2	no	—
168	C₂×C₁₆	—	14	1	no	—
169	C₂×C₁₆	—	14	1	no	—
170	#107	—	2	1	no	—
171	#107	—	2	1	no	—
172	#107	—	2	1	no	—
173	#107	—	2	1	no	—
174	#107	—	14	1	no	—
175	#107	—	14	1	no	—
176	#107	—	14	1	no	—
177	#107	—	14	1	no	—
178	#95	—	28	1	no	—
179	#95	—	28	1	no	—
180	C₂²×C₇	#466	1	1	yes	characteristic
181	C₂²×C₇	#480	1	1	yes	characteristic
182	C₂²×C₇	#480	1	1	yes	characteristic
183	C₂²×C₇	C₂×D₁₆	1	1	yes	characteristic
184	C₂²×C₇	#508	1	1	yes	characteristic
185	C₂²×C₇	#510	1	1	yes	characteristic
186	C₂²×C₇	#510	1	1	yes	characteristic
187	C₂×D₇	—	2	2	no	—
188	C₂×D₇	—	2	2	no	—
189	C₂×D₇	—	2	2	no	—
190	C₂×D₇	—	4	1	no	—
191	C₂×D₇	—	4	1	no	—
192	C₂×D₇	—	4	1	no	—
193	C₂×D₇	—	4	1	no	—
194	C₄×C₇	—	16	1	no	—
195	C₄×C₇	—	16	1	no	—
196	C₄×C₇	#397	1	1	yes	characteristic
197	C₄×C₇	#470	1	1	yes	characteristic
198	C₄×C₇	#485	1	1	yes	characteristic
199	C₄×C₇	#485	1	1	yes	characteristic
200	Dic₇	—	4	1	no	—
201	Dic₇	—	16	1	no	—
202	Dic₇	—	16	1	no	—
203	C₂⁴	—	7	1	no	—
204	C₂²×C₄	—	4	1	no	—
205	C₂²×C₄	—	4	1	no	—
206	C₂²×C₄	—	7	1	no	—
207	C₂²×C₄	—	14	1	no	—
208	C₂²×C₄	—	14	1	no	—
209	C₂²×C₄	—	14	1	no	—
210	C₂²×C₄	—	14	1	no	—
211	C₂²×C₄	—	14	1	no	—
212	C₂²×C₄	—	14	1	no	—
213	C₂²×C₄	—	28	1	no	—
214	C₂²×C₄	—	28	1	no	—
215	C₂²×C₄	D₄×D₇	1	1	yes	characteristic
216	C₂×C₈	—	14	1	no	—
217	C₂×C₈	—	14	1	no	—
218	C₂×C₈	—	14	1	no	—
219	C₂×C₈	—	28	1	no	—
220	C₂×C₈	—	28	1	no	—
221	C₂×C₈	—	28	1	no	—
222	C₂×C₈	—	28	1	no	—
223	C₂×C₈	#1157	1	1	yes	characteristic
224	C₂×C₈	#1157	1	1	yes	characteristic
225	C₂×C₈	#1157	1	1	yes	characteristic
226	C₂×C₈	D₄×D₇	1	1	yes	characteristic
227	C₂×C₈	#1159	1	1	yes	characteristic
228	C₂×C₈	#1161	1	1	yes	characteristic
229	C₄⋊C₄	—	4	1	no	—
230	C₄⋊C₄	—	4	1	no	—
231	C₄⋊C₄	—	4	1	no	—
232	C₄⋊C₄	—	4	1	no	—
233	C₄⋊C₄	—	28	1	no	—
234	C₄⋊C₄	—	28	1	no	—
235	C₄⋊C₄	—	28	1	no	—
236	C₄⋊C₄	—	28	1	no	—
237	C₂²⋊C₄	—	56	2	no	—
238	C₂²⋊C₄	—	56	2	no	—
239	C₁₆	—	4	1	no	—
240	C₁₆	—	14	2	no	—
241	C₂×C₇	#2075	1	1	yes	characteristic
242	C₂×C₇	#2215	1	1	yes	characteristic
243	C₂×C₇	#2219	1	1	yes	characteristic
244	C₂×C₇	#2232	1	1	yes	characteristic
245	C₂×C₇	#2232	1	1	yes	characteristic
246	C₂×C₇	#2237	1	1	yes	characteristic
247	C₂×C₇	#2237	1	1	yes	characteristic
248	D₇	—	4	2	no	—
249	C₂³	—	7	2	no	—
250	C₂³	—	14	1	no	—
251	C₂³	—	14	1	no	—
252	C₂³	—	14	1	no	—
253	C₂³	—	14	1	no	—
254	C₂³	—	14	1	no	—
255	C₂³	—	14	1	no	—
256	C₂³	D₇×D₈	1	1	yes	characteristic
257	C₂×C₄	—	8	1	no	—
258	C₂×C₄	—	8	1	no	—
259	C₂×C₄	—	8	1	no	—
260	C₂×C₄	—	8	1	no	—
261	C₂×C₄	—	8	1	no	—
262	C₂×C₄	—	8	1	no	—
263	C₂×C₄	—	14	1	no	—
264	C₂×C₄	—	14	1	no	—
265	C₂×C₄	—	14	1	no	—
266	C₂×C₄	—	28	1	no	—
267	C₂×C₄	—	28	1	no	—
268	C₂×C₄	—	28	1	no	—
269	C₂×C₄	—	28	1	no	—
270	C₂×C₄	—	56	1	no	—
271	C₂×C₄	—	56	1	no	—
272	C₂×C₄	—	56	1	no	—
273	C₂×C₄	—	56	1	no	—
274	C₂×C₄	—	56	1	no	—
275	C₂×C₄	—	56	1	no	—
276	C₂×C₄	#6446	1	1	yes	characteristic
277	C₂×C₄	#6447	1	1	yes	characteristic
278	C₂×C₄	#6457	1	1	yes	characteristic
279	C₂×C₄	#6478	1	1	yes	characteristic
280	C₂×C₄	#6479	1	1	yes	characteristic
281	C₂×C₄	#6481	1	1	yes	characteristic
282	C₈	—	28	1	no	—
283	C₈	#6439	1	1	yes	characteristic
284	C₈	#6448	1	1	yes	characteristic
285	C₈	#6457	1	1	yes	characteristic
286	C₈	#6461	1	1	yes	characteristic
287	C₇	#13492	1	1	yes	characteristic, complemented, Sylow
288	C₂²	—	14	2	no	—
289	C₂²	—	14	2	no	—
290	C₂²	—	14	2	no	—
291	C₂²	—	28	1	no	—
292	C₂²	—	28	1	no	—
293	C₂²	—	28	1	no	—
294	C₂²	—	28	1	no	—
295	C₂²	#89546	1	1	yes	characteristic
296	C₂²	#89547	1	1	yes	characteristic
297	C₂²	#89654	1	1	yes	characteristic
298	C₂²	D₇×D₁₆	1	1	yes	characteristic
299	C₂²	#89687	1	1	yes	characteristic
300	C₂²	#89688	1	1	yes	characteristic
301	C₂²	#89690	1	1	yes	characteristic
302	C₄	—	16	1	no	—
303	C₄	—	16	1	no	—
304	C₄	—	28	1	no	—
305	C₄	—	112	1	no	—
306	C₄	—	112	1	no	—
307	C₄	#89447	1	1	yes	characteristic
308	C₄	#89587	1	1	yes	characteristic
309	C₄	#89588	1	1	yes	characteristic
310	C₄	#89639	1	1	yes	characteristic
311	C₂	—	28	2	no	—
312	C₂	#11726643	1	1	yes	characteristic
313	C₂	#11726949	1	1	yes	characteristic
314	C₂	#11726953	1	1	yes	characteristic
315	C₂	#11726983	1	1	yes	characteristic
316	C₂	#11726987	1	1	yes	characteristic
317	C₂	#11727084	1	1	yes	characteristic
318	C₂	#11727096	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	D₇×D₈
commutator, cosocle	C₂²×C₇×C₈	C₂³
Frattini	C₂²×C₈	C₂²×D₇
Fitting	C₇×#1380	C₂
socle	C₂³×C₇	C₂×D₈
prime radical	C₂³×D₇	D₈
hypercenter	#1380	D₇
hypocenter	C₇	#13492
LCS	G ▷ C₂²×C₇×C₈ ▷ C₄×C₇ ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ #1380

Complements

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

N	H ≅ G/N	type
4. C₂×#89284	C₂	semidirect
5. C₇×#1380	C₂	semidirect
6. #11725612	C₂	semidirect
7. #11725528	C₂	semidirect
59. #1380	D₇	semidirect
287. C₇	#13492	semidirect

Maximal subgroups

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

subgroup	index
C₂×(C₂×D₇)⋊C₁₆	2
#11728384	2
C₂×#89284	2
C₇×#1380	2
#11725612	2
#11725528	2
#13492	7