Subgroups — #13507298

Order 1792.

498 conjugacy classes of subgroups.

The subgroup lattice has 326 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#11737362	C₂	1	1	yes	maximal, characteristic
2	C₇×Q₈×C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
3	#11728703	C₂	1	1	yes	maximal, characteristic, complemented
4	#11727039	C₂	1	2	yes	maximal
5	D₇×C₄⋊C₁₆	C₂	1	1	yes	maximal, characteristic
6	#11727036	C₂	1	2	yes	maximal, complemented
7	#11727034	C₂	1	1	yes	maximal, characteristic, complemented
8	#11726805	C₂	1	2	yes	maximal
9	C₄×#6372	C₂	1	1	yes	maximal, characteristic
10	#11726802	C₂	1	1	yes	maximal, characteristic, complemented
11	#11726802	C₂	1	2	yes	maximal, complemented
12	#90269	C₄	1	1	yes	characteristic
13	C₇×C₈×Q₈	C₂²	1	1	yes	characteristic
14	#89796	C₄	1	1	yes	characteristic
15	C₂×#6372	C₂²	1	1	yes	characteristic
16	C₂×#6372	C₂²	1	1	yes	characteristic
17	C₂×D₇×C₁₆	C₂²	1	2	yes	—
18	#89611	C₄	1	2	yes	—
19	#89609	C₄	1	1	yes	characteristic
20	#89603	C₄	1	1	yes	characteristic
21	#89600	C₄	1	2	yes	—
22	#89461	C₂²	1	2	yes	—
23	C₄×C₈×D₇	C₂²	1	1	yes	characteristic
24	C₈×Dic₁₄	C₂²	1	1	yes	characteristic
25	C₈×Dic₁₄	C₂²	1	2	yes	—
26	C₇×C₄⋊C₁₆	C₂²	1	1	yes	characteristic
27	C₇×C₄⋊C₁₆	C₂²	1	2	yes	—
28	C₄×C₇×C₁₆	C₂²	1	1	yes	characteristic
29	C₄×C₇×C₁₆	C₂²	1	2	yes	—
30	#89297	C₂²	1	1	yes	characteristic
31	#89297	C₂²	1	1	yes	characteristic
32	#89297	C₂²	1	2	yes	—
33	Dic₇⋊C₁₆	C₂²	1	2	yes	—
34	Dic₇⋊C₁₆	C₂²	1	2	yes	—
35	Dic₇⋊C₁₆	C₂²	1	2	yes	—
36	C₁₆×Dic₇	C₂²	1	2	yes	—
37	C₄⋊(C₇⋊C₁₆)	C₂²	1	1	yes	characteristic
38	C₄⋊(C₇⋊C₁₆)	C₂²	1	2	yes	—
39	#89256	C₂²	1	2	yes	—
40	C₄×C₇⋊C₁₆	C₂²	1	1	yes	characteristic
41	#33476	—	7	1	no	maximal, Sylow
42	C₄×C₇×Q₈	C₂×C₄	1	1	yes	characteristic
43	#6509	C₈	1	1	yes	characteristic
44	#6507	C₈	1	1	yes	characteristic
45	C₂×#1131	C₂×C₄	1	2	yes	—
46	C₂×C₈×D₇	C₂×C₄	1	1	yes	characteristic
47	C₂×C₈×D₇	C₂³	1	1	yes	characteristic
48	#6460	C₈	1	1	yes	characteristic
49	#6459	C₈	1	2	yes	—
50	#6452	C₈	1	2	yes	—
51	#6451	C₈	1	1	yes	characteristic
52	#6435	C₂×C₄	1	2	yes	—
53	C₄²×D₇	C₂×C₄	1	1	yes	characteristic
54	C₄×Dic₁₄	C₂×C₄	1	2	yes	—
55	C₄×Dic₁₄	C₂×C₄	1	1	yes	characteristic
56	C₂×C₇×C₁₆	C₂³	1	2	yes	—
57	C₂×C₇×C₁₆	C₂³	1	1	yes	characteristic
58	C₂×C₇×C₁₆	C₂³	1	1	yes	characteristic
59	C₇×C₄⋊C₈	C₂×C₄	1	2	yes	—
60	C₇×C₄⋊C₈	C₂×C₄	1	1	yes	characteristic
61	C₄×C₇×C₈	C₂³	1	1	yes	characteristic
62	C₄×C₇×C₈	C₂³	1	2	yes	—
63	#6389	C₂×C₄	1	2	yes	—
64	Dic₇⋊C₈	C₂×C₄	1	2	yes	—
65	Dic₇⋊C₈	C₂×C₄	1	2	yes	—
66	Dic₇⋊C₈	C₂×C₄	1	2	yes	—
67	C₈×Dic₇	C₂×C₄	1	1	yes	characteristic
68	C₈×Dic₇	C₂³	1	1	yes	characteristic
69	C₈×Dic₇	C₂³	1	2	yes	—
70	C₂×C₇⋊C₁₆	C₂³	1	2	yes	—
71	C₂×C₇⋊C₁₆	C₂³	1	1	yes	characteristic
72	C₂×C₇⋊C₁₆	C₂³	1	1	yes	characteristic
73	C₄⋊(C₇⋊C₈)	C₂×C₄	1	1	yes	characteristic
74	C₄⋊(C₇⋊C₈)	C₂×C₄	1	2	yes	—
75	#6377	C₂×C₄	1	2	yes	—
76	C₄×C₇⋊C₈	C₂×C₄	1	1	yes	characteristic
77	#6372	—	2	2	no	—
78	#6372	Q₈	1	4	yes	complemented
79	D₇×C₁₆	—	2	4	no	—
80	#2960	—	7	1	no	—
81	#2183	—	7	1	no	—
82	#2183	—	7	2	no	—
83	#2183	—	7	1	no	—
84	#2183	—	7	2	no	—
85	Q₈×C₁₆	—	7	1	no	—
86	Q₈×C₁₆	D₇	1	1	yes	characteristic, complemented
87	#2151	—	7	2	no	—
88	C₂×C₄⋊C₁₆	—	7	1	no	—
89	#2108	—	7	2	no	—
90	C₄×M₃₂	—	7	1	no	—
91	C₂×C₇×Q₈	C₂×C₈	1	1	yes	characteristic
92	C₂×C₄×D₇	C₂×C₈	1	1	yes	characteristic
93	C₂×C₄×D₇	C₂×C₈	1	1	yes	characteristic
94	C₂×C₄×D₇	C₂²×C₄	1	1	yes	characteristic
95	C₂×Dic₁₄	C₂×C₈	1	1	yes	characteristic
96	C₂×Dic₁₄	C₂×C₈	1	2	yes	—
97	C₂×C₇×C₈	C₂²×C₄	1	1	yes	characteristic
98	C₂×C₇×C₈	C₂²×C₄	1	2	yes	—
99	C₂×C₇×C₈	C₂⁴	1	1	yes	characteristic
100	C₇×C₄⋊C₄	C₂×C₈	1	1	yes	characteristic
101	C₇×C₄⋊C₄	C₂×C₈	1	2	yes	—
102	C₄²×C₇	C₂²×C₄	1	1	yes	characteristic
103	C₄²×C₇	C₂²×C₄	1	2	yes	—
104	(C₂×D₇)⋊C₄	C₂×C₈	1	2	yes	—
105	(C₂×D₇)⋊C₄	C₂×C₈	1	2	yes	—
106	C₄⋊Dic₇	C₂×C₈	1	1	yes	characteristic
107	C₄⋊Dic₇	C₂×C₈	1	2	yes	—
108	Dic₇⋊C₄	C₂×C₈	1	2	yes	—
109	Dic₇⋊C₄	C₂×C₈	1	2	yes	—
110	Dic₇⋊C₄	C₂×C₈	1	2	yes	—
111	Dic₇⋊C₄	C₂×C₈	1	2	yes	—
112	Dic₇⋊C₄	C₂×C₈	1	2	yes	—
113	C₄×Dic₇	C₂×C₈	1	1	yes	characteristic
114	C₄×Dic₇	C₂×C₈	1	1	yes	characteristic
115	C₄×Dic₇	C₂²×C₄	1	1	yes	characteristic
116	C₄×Dic₇	C₂²×C₄	1	2	yes	—
117	C₂×C₇⋊C₈	C₂²×C₄	1	1	yes	characteristic
118	C₂×C₇⋊C₈	C₂²×C₄	1	2	yes	—
119	C₂×C₇⋊C₈	C₂²×C₄	1	1	yes	characteristic
120	#1131	—	2	4	no	—
121	C₈×D₇	—	2	2	no	—
122	C₈×D₇	C₂×Q₈	1	2	yes	—
123	C₈×D₇	#41	1	2	yes	—
124	C₇×C₁₆	—	2	2	no	—
125	C₇×C₁₆	—	2	1	no	—
126	C₇×C₁₆	C₂×Q₈	1	2	yes	—
127	C₇⋊C₁₆	—	2	2	no	—
128	C₇⋊C₁₆	—	2	1	no	—
129	C₇⋊C₁₆	C₂×Q₈	1	2	yes	—
130	#533	—	7	1	no	—
131	C₂×M₃₂	—	7	1	no	—
132	C₂×M₃₂	—	7	1	no	—
133	C₂²×C₁₆	—	7	2	no	—
134	#446	—	7	1	no	—
135	#446	—	7	2	no	—
136	#446	—	7	1	no	—
137	C₈×Q₈	—	7	1	no	—
138	C₈×Q₈	—	7	2	no	—
139	C₈×Q₈	C₂×D₇	1	1	yes	characteristic
140	#424	—	7	2	no	—
141	#423	—	7	1	no	—
142	#405	—	7	2	no	—
143	C₂×C₄×C₈	—	7	1	no	—
144	C₄⋊C₁₆	—	7	2	no	—
145	C₄⋊C₁₆	—	7	2	no	—
146	C₄⋊C₁₆	—	7	2	no	—
147	C₄⋊C₁₆	—	7	1	no	—
148	C₄⋊C₁₆	—	7	2	no	—
149	C₄⋊C₁₆	C₂×D₇	1	1	yes	characteristic
150	C₄⋊C₁₆	C₂×D₇	1	2	yes	—
151	#346	—	7	1	no	—
152	#346	—	7	1	no	—
153	#346	—	7	2	no	—
154	#346	—	7	2	no	—
155	C₄×C₁₆	—	7	2	no	—
156	C₄×C₁₆	—	7	1	no	—
157	C₄×C₁₆	C₂×D₇	1	1	yes	characteristic
158	C₄×C₁₆	C₂×D₇	1	2	yes	—
159	C₂²×D₇	C₂²×C₈	1	1	yes	characteristic
160	C₇×Q₈	—	2	2	no	—
161	C₂×C₄×C₇	C₂²×C₈	1	1	yes	characteristic
162	C₂×C₄×C₇	C₂²×C₈	1	2	yes	—
163	C₂×C₄×C₇	C₂²×C₈	1	2	yes	—
164	C₂×C₄×C₇	C₂²×C₈	1	1	yes	characteristic
165	C₂×C₄×C₇	C₂³×C₄	1	1	yes	characteristic
166	C₂×Dic₇	C₂²×C₈	1	1	yes	characteristic
167	C₂×Dic₇	C₂²×C₈	1	1	yes	characteristic
168	C₂×Dic₇	C₂²×C₈	1	2	yes	—
169	C₂×Dic₇	C₂²×C₈	1	2	yes	—
170	C₂×Dic₇	C₂²×C₈	1	1	yes	characteristic
171	C₄×D₇	—	2	2	no	—
172	C₄×D₇	—	4	1	no	—
173	C₄×D₇	C₄×Q₈	1	2	yes	—
174	C₄×D₇	#131	1	2	yes	—
175	Dic₁₄	—	2	2	no	—
176	Dic₁₄	—	4	2	no	—
177	C₇×C₈	—	2	1	no	—
178	C₇×C₈	—	2	2	no	—
179	C₇×C₈	C₂²×Q₈	1	1	yes	characteristic
180	C₇×C₈	C₂×#41	1	1	yes	characteristic
181	C₇⋊C₈	—	2	1	no	—
182	C₇⋊C₈	—	2	2	no	—
183	C₇⋊C₈	C₄×Q₈	1	2	yes	—
184	C₂×M₁₆	—	7	2	no	—
185	C₂²×C₈	—	7	1	no	—
186	C₂²×C₈	—	7	1	no	—
187	C₄⋊Q₈	—	7	1	no	—
188	C₄⋊Q₈	—	7	1	no	—
189	#125	—	7	2	no	—
190	C₂²⋊Q₈	—	7	1	no	—
191	C₂²⋊Q₈	—	7	2	no	—
192	C₂²⋊Q₈	—	7	1	no	—
193	C₄×Q₈	—	7	2	no	—
194	C₄×Q₈	—	7	1	no	—
195	C₄×Q₈	C₄×D₇	1	1	yes	characteristic
196	#117	—	7	2	no	—
197	C₂×C₄²	—	7	1	no	—
198	M₃₂	—	7	4	no	—
199	M₃₂	—	14	2	no	—
200	C₂×C₁₆	—	7	1	no	—
201	C₂×C₁₆	—	7	1	no	—
202	C₂×C₁₆	—	7	2	no	—
203	C₂×C₁₆	—	14	4	no	—
204	C₂×C₁₆	C₂²×D₇	1	2	yes	—
205	C₂×C₁₆	C₂²×D₇	1	1	yes	characteristic
206	C₂×C₁₆	C₂²×D₇	1	1	yes	characteristic
207	C₄⋊C₈	—	7	2	no	—
208	C₄⋊C₈	—	7	2	no	—
209	C₄⋊C₈	—	7	2	no	—
210	C₄⋊C₈	—	7	1	no	—
211	C₄⋊C₈	—	7	2	no	—
212	C₄⋊C₈	C₄×D₇	1	1	yes	characteristic
213	C₄⋊C₈	C₄×D₇	1	2	yes	—
214	#97	—	7	2	no	—
215	#97	—	7	2	no	—
216	C₄×C₈	—	7	1	no	—
217	C₄×C₈	—	7	2	no	—
218	C₄×C₈	—	7	1	no	—
219	C₄×C₈	—	7	1	no	—
220	C₄×C₈	C₂²×D₇	1	1	yes	characteristic
221	C₄×C₈	C₂²×D₇	1	2	yes	—
222	C₂²×C₇	C₂³×C₈	1	1	yes	characteristic
223	C₂×D₇	—	2	1	no	—
224	C₂×D₇	—	2	1	no	—
225	C₂×D₇	C₈×Q₈	1	2	yes	—
226	C₄×C₇	—	2	2	no	—
227	C₄×C₇	—	2	2	no	—
228	C₄×C₇	—	2	1	no	—
229	C₄×C₇	#504	1	2	yes	—
230	C₄×C₇	C₂×C₄×Q₈	1	1	yes	characteristic
231	C₄×C₇	C₂×#131	1	1	yes	characteristic
232	Dic₇	—	2	2	no	—
233	Dic₇	—	2	1	no	—
234	Dic₇	—	2	1	no	—
235	Dic₇	—	2	2	no	—
236	Dic₇	C₈×Q₈	1	2	yes	—
237	C₂×Q₈	—	7	1	no	—
238	C₂×Q₈	—	7	2	no	—
239	C₂×Q₈	C₈×D₇	1	1	yes	characteristic
240	C₂²×C₄	—	7	1	no	—
241	C₂²×C₄	—	7	1	no	—
242	C₂²×C₄	—	7	1	no	—
243	M₁₆	—	14	4	no	—
244	C₂×C₈	—	7	1	no	—
245	C₂×C₈	—	7	2	no	—
246	C₂×C₈	—	7	2	no	—
247	C₂×C₈	—	7	1	no	—
248	C₂×C₈	—	7	2	no	—
249	C₂×C₈	—	14	2	no	—
250	C₂×C₈	C₂×C₄×D₇	1	1	yes	characteristic
251	C₂×C₈	C₂×C₄×D₇	1	2	yes	—
252	C₂×C₈	C₂³×D₇	1	1	yes	characteristic
253	C₄⋊C₄	—	7	2	no	—
254	C₄⋊C₄	—	7	2	no	—
255	C₄⋊C₄	—	7	2	no	—
256	C₄⋊C₄	—	7	2	no	—
257	C₄⋊C₄	—	7	2	no	—
258	C₄⋊C₄	—	7	2	no	—
259	C₄⋊C₄	—	7	1	no	—
260	C₄⋊C₄	C₈×D₇	1	2	yes	—
261	C₄⋊C₄	C₈×D₇	1	1	yes	characteristic
262	C₂²⋊C₄	—	7	2	no	—
263	C₂²⋊C₄	—	7	2	no	—
264	C₄²	—	7	1	no	—
265	C₄²	—	7	1	no	—
266	C₄²	—	7	1	no	—
267	C₄²	—	7	2	no	—
268	C₄²	C₂×C₄×D₇	1	1	yes	characteristic
269	C₄²	C₂×C₄×D₇	1	2	yes	—
270	C₁₆	—	2	2	no	—
271	C₁₆	—	2	1	no	—
272	C₁₆	—	7	2	no	—
273	C₁₆	—	14	2	no	—
274	C₁₆	—	14	1	no	—
275	C₁₆	Q₈×D₇	1	2	yes	—
276	C₂×C₇	C₂×C₈×Q₈	1	1	yes	characteristic
277	C₂×C₇	C₂×#504	1	1	yes	characteristic
278	C₂×C₇	#3407	1	1	yes	characteristic
279	D₇	—	2	2	no	—
280	C₂³	—	7	1	no	—
281	Q₈	—	2	2	no	—
282	Q₈	—	14	2	no	—
283	Q₈	—	28	2	no	—
284	C₂×C₄	—	7	1	no	—
285	C₂×C₄	—	7	2	no	—
286	C₂×C₄	—	7	2	no	—
287	C₂×C₄	—	7	1	no	—
288	C₂×C₄	—	7	1	no	—
289	C₂×C₄	—	7	2	no	—
290	C₂×C₄	—	7	2	no	—
291	C₂×C₄	—	14	2	no	—
292	C₂×C₄	—	28	1	no	—
293	C₂×C₄	C₂×C₈×D₇	1	1	yes	characteristic
294	C₂×C₄	C₂×C₈×D₇	1	2	yes	—
295	C₂×C₄	C₂×C₈×D₇	1	2	yes	—
296	C₂×C₄	C₂×C₈×D₇	1	1	yes	characteristic
297	C₂×C₄	C₂²×C₄×D₇	1	1	yes	characteristic
298	C₈	—	2	1	no	—
299	C₈	—	2	2	no	—
300	C₈	—	7	2	no	—
301	C₈	—	14	1	no	—
302	C₈	—	14	2	no	—
303	C₈	C₂×Q₈×D₇	1	1	yes	characteristic
304	C₈	D₇×#41	1	1	yes	characteristic
305	C₇	#33476	1	1	yes	characteristic, complemented, Sylow
306	C₂²	—	7	2	no	—
307	C₂²	—	14	1	no	—
308	C₂²	—	14	1	no	—
309	C₂²	C₂²×C₈×D₇	1	1	yes	characteristic
310	C₄	—	2	2	no	—
311	C₄	—	2	1	no	—
312	C₄	—	2	2	no	—
313	C₄	—	7	2	no	—
314	C₄	—	14	1	no	—
315	C₄	—	14	2	no	—
316	C₄	—	14	1	no	—
317	C₄	—	14	2	no	—
318	C₄	#89673	1	2	yes	—
319	C₄	C₄×Q₈×D₇	1	1	yes	characteristic
320	C₄	D₇×#131	1	1	yes	characteristic
321	C₂	—	14	2	no	—
322	C₂	C₈×Q₈×D₇	1	1	yes	characteristic
323	C₂	C₂×#89673	1	1	yes	characteristic
324	C₂	#11739457	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₈	C₂³×D₇
commutator, socle	C₂²×C₇	C₂³×C₈
Fitting	C₇×Q₈×C₁₆	C₂
prime radical	C₂²×D₇	C₂²×C₈
cosocle	C₂×C₇×C₈	C₂⁴
hypercenter	Q₈×C₁₆	D₇
hypocenter	C₇	#33476
LCS	G ▷ C₂²×C₇ ▷ C₇
UCS	C₁ ◁ C₂×C₈ ◁ Q₈×C₁₆

Complements

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

N	H ≅ G/N	type
2. C₇×Q₈×C₁₆	C₂	semidirect
3. #11728703	C₂	semidirect
6. #11727036	C₂	semidirect
7. #11727034	C₂	semidirect
10. #11726802	C₂	semidirect
11. #11726802	C₂	semidirect
78. #6372	Q₈	semidirect
86. Q₈×C₁₆	D₇	semidirect
305. C₇	#33476	semidirect

Maximal subgroups

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

subgroup	index
#11737362	2
C₇×Q₈×C₁₆	2
#11728703	2
#11727039	2
D₇×C₄⋊C₁₆	2
#11727036	2
#11727034	2
#11726805	2
C₄×#6372	2
#11726802	2
#33476	7