Subgroups — #13325485

Order 1792.

626 conjugacy classes of subgroups.

The subgroup lattice has 330 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×C₇×Q₁₆	C₂	1	1	yes	maximal, characteristic
2	#11739882	C₂	1	2	yes	maximal
3	#11729377	C₂	1	2	yes	maximal
4	#11728659	C₂	1	2	yes	maximal, complemented
5	#11728658	C₂	1	4	yes	maximal, complemented
6	Q₁₆×C₇⋊C₈	C₂	1	2	yes	maximal, complemented
7	#11727486	C₂	1	1	yes	maximal, characteristic
8	C₈×#1136	C₂	1	1	yes	maximal, characteristic
9	C₂²×C₇×Q₁₆	C₄	1	1	yes	characteristic
10	C₂×C₄×C₇×Q₈	C₂²	1	2	yes	—
11	C₄×C₇×Q₁₆	C₄	1	2	yes	—
12	C₄×C₇×Q₁₆	C₄	1	1	yes	characteristic
13	C₄×C₇×Q₁₆	C₄	1	1	yes	characteristic
14	C₄×C₇×Q₁₆	C₂²	1	1	yes	characteristic
15	C₄×C₇×Q₁₆	C₂²	1	2	yes	—
16	C₄×C₇×Q₁₆	C₂²	1	1	yes	characteristic
17	C₂×C₇×#107	C₄	1	1	yes	characteristic
18	C₂×C₇×#103	C₄	1	2	yes	—
19	C₂×C₄×C₇×C₈	C₂²	1	1	yes	characteristic
20	#89796	—	2	4	no	—
21	#89796	C₂²	1	4	yes	—
22	Q₈×C₇⋊C₈	C₂²	1	4	yes	—
23	#89698	—	2	2	no	—
24	#89698	C₂²	1	2	yes	—
25	#89697	—	2	2	no	—
26	#89697	C₂²	1	1	yes	characteristic
27	C₄×#1136	C₂²	1	1	yes	characteristic
28	Q₈⋊(C₇⋊C₈)	C₂²	1	4	yes	—
29	Q₈⋊(C₇⋊C₈)	C₂²	1	4	yes	—
30	#89253	C₂²	1	2	yes	—
31	#89252	C₂²	1	2	yes	—
32	#89249	C₂²	1	2	yes	—
33	C₈×C₇⋊C₈	C₂²	1	2	yes	—
34	#18058	—	7	1	no	maximal, Sylow
35	C₂²×C₇×Q₈	C₂×C₄	1	2	yes	—
36	C₂×C₇×Q₁₆	—	2	4	no	—
37	C₂×C₇×Q₁₆	C₂×C₄	1	2	yes	—
38	C₂×C₇×Q₁₆	C₂×C₄	1	1	yes	characteristic
39	C₂×C₇×Q₁₆	C₂×C₄	1	1	yes	characteristic
40	C₂²×C₇×C₈	C₂×C₄	1	1	yes	characteristic
41	C₂²×C₇×C₈	C₂×C₄	1	1	yes	characteristic
42	C₄×C₇×Q₈	—	2	2	no	—
43	C₄×C₇×Q₈	—	2	2	no	—
44	C₄×C₇×Q₈	C₂×C₄	1	2	yes	—
45	C₄×C₇×Q₈	C₂×C₄	1	2	yes	—
46	C₄×C₇×Q₈	C₂³	1	2	yes	—
47	C₄×C₇×Q₈	C₂³	1	2	yes	—
48	C₂×C₇×C₄⋊C₄	—	2	2	no	—
49	C₂×C₇×C₄⋊C₄	C₂×C₄	1	2	yes	—
50	C₂×C₄²×C₇	—	2	2	no	—
51	C₂×C₄²×C₇	C₂³	1	1	yes	characteristic
52	C₂×#1136	—	2	1	no	—
53	C₇×#107	C₂×C₄	1	2	yes	—
54	C₇×#107	C₂×C₄	1	1	yes	characteristic
55	C₇×#107	C₂×C₄	1	1	yes	characteristic
56	C₇×#103	C₂×C₄	1	2	yes	—
57	C₇×#103	C₂×C₄	1	2	yes	—
58	C₇×#103	C₂×C₄	1	2	yes	—
59	C₇×#103	C₂×C₄	1	2	yes	—
60	C₄×C₇×C₈	C₂×C₄	1	1	yes	characteristic
61	C₄×C₇×C₈	C₂×C₄	1	1	yes	characteristic
62	C₄×C₇×C₈	C₂³	1	1	yes	characteristic
63	C₄×C₇×C₈	C₂³	1	1	yes	characteristic
64	C₂²⋊(C₇⋊C₈)	—	2	2	no	—
65	C₂²⋊(C₇⋊C₈)	D₄	1	1	yes	characteristic
66	C₂²⋊(C₇⋊C₈)	D₄	1	1	yes	characteristic
67	C₄⋊(C₇⋊C₈)	—	2	4	no	—
68	C₄⋊(C₇⋊C₈)	—	2	4	no	—
69	C₄⋊(C₇⋊C₈)	D₄	1	2	yes	—
70	C₄⋊(C₇⋊C₈)	C₂³	1	4	yes	—
71	#6377	—	2	2	no	—
72	#6377	—	2	1	no	—
73	#6377	—	2	4	no	—
74	C₄×C₇⋊C₈	—	2	4	no	—
75	C₄×C₇⋊C₈	C₂³	1	2	yes	—
76	C₄×C₇⋊C₈	C₂³	1	2	yes	—
77	#2984	—	7	2	no	—
78	C₂×C₄×Q₁₆	D₇	1	1	yes	characteristic
79	#1591	—	7	2	no	—
80	#1587	—	7	4	no	—
81	C₈×Q₁₆	—	7	2	no	—
82	#1568	—	7	1	no	—
83	#1491	—	7	2	no	—
84	C₈×M₁₆	—	7	1	no	—
85	C₂×C₇×Q₈	—	2	2	no	—
86	C₂×C₇×Q₈	—	2	4	no	—
87	C₂×C₇×Q₈	—	4	2	no	—
88	C₂×C₇×Q₈	C₂²×C₄	1	2	yes	—
89	C₂×C₇×Q₈	C₂²×C₄	1	2	yes	—
90	C₂²×C₄×C₇	—	2	2	no	—
91	C₂²×C₄×C₇	—	2	2	no	—
92	C₂²×C₄×C₇	C₂²×C₄	1	1	yes	characteristic
93	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
94	C₂²×C₄×C₇	#41	1	1	yes	characteristic
95	C₇×Q₁₆	—	2	4	no	—
96	C₇×Q₁₆	—	2	2	no	—
97	C₇×Q₁₆	M₁₆	1	4	yes	complemented
98	C₂×C₇×C₈	—	2	2	no	—
99	C₂×C₇×C₈	—	4	1	no	—
100	C₂×C₇×C₈	C₂²×C₄	1	1	yes	characteristic
101	C₂×C₇×C₈	C₂²×C₄	1	1	yes	characteristic
102	C₂×C₇×C₈	C₂²×C₄	1	1	yes	characteristic
103	C₂×C₇×C₈	C₂²×C₄	1	1	yes	characteristic
104	C₇×C₄⋊C₄	—	2	2	no	—
105	C₇×C₄⋊C₄	—	2	2	no	—
106	C₇×C₄⋊C₄	—	2	2	no	—
107	C₇×C₄⋊C₄	—	2	2	no	—
108	C₇×C₄⋊C₄	—	2	2	no	—
109	C₇×C₄⋊C₄	C₂²×C₄	1	2	yes	—
110	C₇×C₄⋊C₄	C₂²×C₄	1	2	yes	—
111	C₄²×C₇	—	2	2	no	—
112	C₄²×C₇	—	2	2	no	—
113	C₄²×C₇	—	2	2	no	—
114	C₄²×C₇	—	2	2	no	—
115	C₄²×C₇	C₂²×C₄	1	1	yes	characteristic
116	C₄²×C₇	C₂×D₄	1	1	yes	characteristic
117	C₄²×C₇	#41	1	1	yes	characteristic
118	C₄²×C₇	C₂⁴	1	1	yes	characteristic
119	#1136	—	2	4	no	—
120	C₂×C₇⋊C₈	—	2	2	no	—
121	C₂×C₇⋊C₈	—	2	4	no	—
122	C₂×C₇⋊C₈	C₂×D₄	1	2	yes	—
123	C₂×C₇⋊C₈	C₂×D₄	1	2	yes	—
124	C₂²×Q₁₆	Dic₇	1	1	yes	characteristic
125	C₂×C₄×Q₈	C₂×D₇	1	2	yes	—
126	#446	—	7	4	no	—
127	#446	—	14	4	no	—
128	C₈×Q₈	—	7	4	no	—
129	C₄×Q₁₆	Dic₇	1	2	yes	—
130	C₄×Q₁₆	Dic₇	1	1	yes	characteristic
131	C₄×Q₁₆	Dic₇	1	1	yes	characteristic
132	C₄×Q₁₆	C₂×D₇	1	1	yes	characteristic
133	C₄×Q₁₆	C₂×D₇	1	2	yes	—
134	C₄×Q₁₆	C₂×D₇	1	1	yes	characteristic
135	#432	—	7	2	no	—
136	#432	—	14	2	no	—
137	#431	—	7	1	no	—
138	#431	—	14	2	no	—
139	C₂×#107	Dic₇	1	1	yes	characteristic
140	C₂×#103	Dic₇	1	2	yes	—
141	C₄×M₁₆	—	7	1	no	—
142	C₂×C₄×C₈	C₂×D₇	1	1	yes	characteristic
143	#335	—	7	2	no	—
144	#334	—	7	2	no	—
145	Q₈⋊C₈	—	7	4	no	—
146	Q₈⋊C₈	—	7	4	no	—
147	#322	—	7	2	no	—
148	C₈²	—	7	2	no	—
149	C₂³×C₇	C₄×D₄	1	1	yes	characteristic
150	C₇×Q₈	—	2	2	no	—
151	C₇×Q₈	—	2	2	no	—
152	C₇×Q₈	—	2	2	no	—
153	C₇×Q₈	—	4	4	no	—
154	C₇×Q₈	C₂×M₁₆	1	4	yes	—
155	C₂×C₄×C₇	—	2	2	no	—
156	C₂×C₄×C₇	—	2	2	no	—
157	C₂×C₄×C₇	—	2	1	no	—
158	C₂×C₄×C₇	—	2	1	no	—
159	C₂×C₄×C₇	—	2	2	no	—
160	C₂×C₄×C₇	—	2	2	no	—
161	C₂×C₄×C₇	—	2	2	no	—
162	C₂×C₄×C₇	—	2	2	no	—
163	C₂×C₄×C₇	—	4	4	no	—
164	C₂×C₄×C₇	—	4	4	no	—
165	C₂×C₄×C₇	C₄×D₄	1	1	yes	characteristic
166	C₂×C₄×C₇	C₄×D₄	1	1	yes	characteristic
167	C₂×C₄×C₇	C₄×D₄	1	1	yes	characteristic
168	C₂×C₄×C₇	C₂³×C₄	1	1	yes	characteristic
169	C₂×C₄×C₇	C₂²×D₄	1	1	yes	characteristic
170	C₂×C₄×C₇	C₂×#41	1	1	yes	characteristic
171	C₇×C₈	—	2	1	no	—
172	C₇×C₈	—	2	1	no	—
173	C₇×C₈	—	2	1	no	—
174	C₇×C₈	C₂×M₁₆	1	2	yes	—
175	C₇⋊C₈	—	2	4	no	—
176	C₇⋊C₈	—	2	2	no	—
177	C₇⋊C₈	—	2	2	no	—
178	C₇⋊C₈	—	4	4	no	—
179	C₂²×Q₈	C₂×Dic₇	1	2	yes	—
180	C₂×Q₁₆	—	2	4	no	—
181	C₂×Q₁₆	C₂×Dic₇	1	2	yes	—
182	C₂×Q₁₆	C₂×Dic₇	1	1	yes	characteristic
183	C₂×Q₁₆	C₂×Dic₇	1	1	yes	characteristic
184	C₂×M₁₆	—	14	1	no	—
185	C₂²×C₈	C₂×Dic₇	1	1	yes	characteristic
186	C₂²×C₈	C₂×Dic₇	1	1	yes	characteristic
187	C₄×Q₈	—	2	2	no	—
188	C₄×Q₈	—	2	2	no	—
189	C₄×Q₈	C₂×Dic₇	1	2	yes	—
190	C₄×Q₈	C₂×Dic₇	1	2	yes	—
191	C₄×Q₈	C₂²×D₇	1	2	yes	—
192	C₄×Q₈	C₂²×D₇	1	2	yes	—
193	C₂×C₄⋊C₄	—	2	2	no	—
194	C₂×C₄⋊C₄	C₂×Dic₇	1	2	yes	—
195	C₂×C₄²	—	2	2	no	—
196	C₂×C₄²	C₂²×D₇	1	1	yes	characteristic
197	#107	C₂×Dic₇	1	2	yes	—
198	#107	C₂×Dic₇	1	1	yes	characteristic
199	#107	C₂×Dic₇	1	1	yes	characteristic
200	C₄⋊C₈	—	7	2	no	—
201	C₄⋊C₈	—	7	4	no	—
202	C₄⋊C₈	—	14	4	no	—
203	C₄⋊C₈	—	14	4	no	—
204	#103	C₂×Dic₇	1	2	yes	—
205	#103	C₂×Dic₇	1	2	yes	—
206	#103	C₂×Dic₇	1	2	yes	—
207	#103	C₂×Dic₇	1	2	yes	—
208	C₂²⋊C₈	—	7	1	no	—
209	C₂²⋊C₈	—	7	1	no	—
210	C₂²⋊C₈	—	14	2	no	—
211	#97	—	14	4	no	—
212	#97	—	14	2	no	—
213	#97	—	14	1	no	—
214	C₄×C₈	—	7	2	no	—
215	C₄×C₈	—	7	2	no	—
216	C₄×C₈	—	14	4	no	—
217	C₄×C₈	C₂×Dic₇	1	1	yes	characteristic
218	C₄×C₈	C₂×Dic₇	1	1	yes	characteristic
219	C₄×C₈	C₂²×D₇	1	1	yes	characteristic
220	C₄×C₈	C₂²×D₇	1	1	yes	characteristic
221	C₂²×C₇	—	2	1	no	—
222	C₂²×C₇	—	2	1	no	—
223	C₂²×C₇	#443	1	2	yes	—
224	C₂²×C₇	C₂×C₄×D₄	1	1	yes	characteristic
225	C₄×C₇	—	2	1	no	—
226	C₄×C₇	—	2	4	no	—
227	C₄×C₇	—	2	1	no	—
228	C₄×C₇	—	2	1	no	—
229	C₄×C₇	—	2	1	no	—
230	C₄×C₇	—	4	2	no	—
231	C₄×C₇	—	4	2	no	—
232	C₄×C₇	—	4	2	no	—
233	C₄×C₇	C₂²×M₁₆	1	1	yes	characteristic
234	C₄×C₇	#568	1	1	yes	characteristic
235	C₄×C₇	#578	1	2	yes	—
236	C₂×Q₈	—	2	2	no	—
237	C₂×Q₈	—	2	4	no	—
238	C₂×Q₈	—	4	2	no	—
239	C₂×Q₈	C₂²×Dic₇	1	2	yes	—
240	C₂×Q₈	C₂²×Dic₇	1	2	yes	—
241	C₂²×C₄	—	2	2	no	—
242	C₂²×C₄	—	2	2	no	—
243	C₂²×C₄	D₄×D₇	1	1	yes	characteristic
244	C₂²×C₄	#1159	1	1	yes	characteristic
245	C₂²×C₄	C₂²×Dic₇	1	1	yes	characteristic
246	Q₁₆	—	2	4	no	—
247	Q₁₆	—	2	2	no	—
248	Q₁₆	#1136	1	4	yes	complemented
249	M₁₆	—	14	4	no	—
250	C₂×C₈	—	2	2	no	—
251	C₂×C₈	—	4	1	no	—
252	C₂×C₈	—	7	2	no	—
253	C₂×C₈	—	7	2	no	—
254	C₂×C₈	—	14	2	no	—
255	C₂×C₈	—	14	4	no	—
256	C₂×C₈	C₂²×Dic₇	1	1	yes	characteristic
257	C₂×C₈	C₂²×Dic₇	1	1	yes	characteristic
258	C₂×C₈	C₂²×Dic₇	1	1	yes	characteristic
259	C₂×C₈	C₂²×Dic₇	1	1	yes	characteristic
260	C₄⋊C₄	—	2	2	no	—
261	C₄⋊C₄	—	2	2	no	—
262	C₄⋊C₄	—	2	2	no	—
263	C₄⋊C₄	—	2	2	no	—
264	C₄⋊C₄	—	2	2	no	—
265	C₄⋊C₄	C₂²×Dic₇	1	2	yes	—
266	C₄⋊C₄	C₂²×Dic₇	1	2	yes	—
267	C₄²	—	2	2	no	—
268	C₄²	—	2	2	no	—
269	C₄²	—	2	2	no	—
270	C₄²	—	2	2	no	—
271	C₄²	D₄×D₇	1	1	yes	characteristic
272	C₄²	#1159	1	1	yes	characteristic
273	C₄²	C₂²×Dic₇	1	1	yes	characteristic
274	C₄²	C₂³×D₇	1	1	yes	characteristic
275	C₂×C₇	—	2	2	no	—
276	C₂×C₇	D₄×M₁₆	1	1	yes	characteristic
277	C₂×C₇	#2950	1	1	yes	characteristic
278	C₂×C₇	C₂×#443	1	1	yes	characteristic
279	C₂³	D₄×Dic₇	1	1	yes	characteristic
280	Q₈	—	2	2	no	—
281	Q₈	—	2	2	no	—
282	Q₈	—	2	2	no	—
283	Q₈	—	4	4	no	—
284	Q₈	C₂×#1136	1	4	yes	—
285	C₂×C₄	—	2	2	no	—
286	C₂×C₄	—	2	2	no	—
287	C₂×C₄	—	2	2	no	—
288	C₂×C₄	—	2	1	no	—
289	C₂×C₄	—	2	1	no	—
290	C₂×C₄	—	2	2	no	—
291	C₂×C₄	—	2	2	no	—
292	C₂×C₄	—	2	2	no	—
293	C₂×C₄	—	4	4	no	—
294	C₂×C₄	—	4	4	no	—
295	C₂×C₄	D₄×Dic₇	1	1	yes	characteristic
296	C₂×C₄	D₄×Dic₇	1	1	yes	characteristic
297	C₂×C₄	D₄×Dic₇	1	1	yes	characteristic
298	C₂×C₄	C₂×D₄×D₇	1	1	yes	characteristic
299	C₂×C₄	C₂×#1159	1	1	yes	characteristic
300	C₂×C₄	C₂³×Dic₇	1	1	yes	characteristic
301	C₈	—	2	1	no	—
302	C₈	—	2	1	no	—
303	C₈	—	2	1	no	—
304	C₈	—	14	4	no	—
305	C₈	—	14	2	no	—
306	C₈	—	14	2	no	—
307	C₈	—	28	4	no	—
308	C₈	C₂×#1136	1	2	yes	—
309	C₇	#18058	1	1	yes	characteristic, complemented, Sylow
310	C₂²	—	2	1	no	—
311	C₂²	—	2	1	no	—
312	C₂²	#89968	1	2	yes	—
313	C₂²	C₂×D₄×Dic₇	1	1	yes	characteristic
314	C₄	—	2	1	no	—
315	C₄	—	2	4	no	—
316	C₄	—	2	1	no	—
317	C₄	—	2	1	no	—
318	C₄	—	2	1	no	—
319	C₄	—	4	2	no	—
320	C₄	—	4	2	no	—
321	C₄	—	4	2	no	—
322	C₄	#90457	1	2	yes	—
323	C₄	C₂²×#1136	1	1	yes	characteristic
324	C₄	#90510	1	1	yes	characteristic
325	C₂	—	2	2	no	—
326	C₂	D₄×#1136	1	1	yes	characteristic
327	C₂	#11741319	1	1	yes	characteristic
328	C₂	C₂×#89968	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×D₄×D₇
commutator	C₂×C₄×C₇	C₂³×C₄
Frattini	C₄²	C₂³×D₇
Fitting	C₂×C₄×C₇×Q₁₆	C₂
socle	C₂²×C₇	C₂×C₄×D₄
prime radical	C₂³×C₇	C₄×D₄
cosocle	C₄²×C₇	C₂⁴
hypercenter	C₂×C₄×Q₁₆	D₇
hypocenter	C₇	#18058
LCS	G ▷ C₂×C₄×C₇ ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₄×Q₁₆

Complements

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

N	H ≅ G/N	type
4. #11728659	C₂	semidirect
5. #11728658	C₂	semidirect
6. Q₁₆×C₇⋊C₈	C₂	semidirect
97. C₇×Q₁₆	M₁₆	semidirect
248. Q₁₆	#1136	semidirect
309. C₇	#18058	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄×C₇×Q₁₆	2
#11739882	2
#11729377	2
#11728659	2
#11728658	2
Q₁₆×C₇⋊C₈	2
#11727486	2
C₈×#1136	2
#18058	7