Subgroups — #13322914

Order 1792.

648 conjugacy classes of subgroups.

The subgroup lattice has 360 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₇×#2920	C₂	1	1	yes	maximal, characteristic
2	C₂×#89698	C₂	1	1	yes	maximal, characteristic
3	C₂×#89697	C₂	1	1	yes	maximal, characteristic
4	#11728506	C₂	1	4	yes	maximal, complemented
5	#11728505	C₂	1	4	yes	maximal, complemented
6	#11727604	C₂	1	4	yes	maximal, complemented
7	C₂²×C₄²×C₇	C₂²	1	1	yes	characteristic
8	C₇×#432	C₄	1	2	yes	—
9	C₇×#432	C₄	1	2	yes	—
10	C₇×#431	C₂²	1	2	yes	—
11	C₇×#431	C₂²	1	2	yes	—
12	C₇×#423	C₄	1	2	yes	—
13	C₇×#407	C₄	1	2	yes	—
14	C₄×C₇×M₁₆	C₂²	1	2	yes	—
15	C₂×C₂²⋊(C₇⋊C₈)	—	2	1	no	—
16	C₂×C₂²⋊(C₇⋊C₈)	—	2	1	no	—
17	#89698	—	2	1	no	—
18	#89698	—	2	1	no	—
19	#89698	C₂²	1	2	yes	—
20	#89698	C₂²	1	2	yes	—
21	#89697	—	2	1	no	—
22	#89697	—	2	1	no	—
23	#89697	C₂²	1	2	yes	—
24	#89697	C₂²	1	2	yes	—
25	C₂×C₄⋊(C₇⋊C₈)	C₂²	1	2	yes	—
26	C₂×C₄⋊(C₇⋊C₈)	C₂²	1	2	yes	—
27	C₂×#6377	—	2	1	no	—
28	C₂×C₄×C₇⋊C₈	—	2	1	no	—
29	#89253	C₂²	1	8	yes	complemented
30	#89252	C₂²	1	8	yes	complemented
31	#17810	—	7	1	no	maximal, Sylow
32	C₂³×C₄×C₇	C₂×C₄	1	1	yes	characteristic
33	C₂³×C₄×C₇	D₄	1	1	yes	characteristic
34	C₂³×C₄×C₇	Q₈	1	1	yes	characteristic
35	C₂×C₇×M₁₆	C₂×C₄	1	4	yes	—
36	C₂×C₄²×C₇	C₂×C₄	1	1	yes	characteristic
37	C₂×C₄²×C₇	C₂×C₄	1	1	yes	characteristic
38	C₂×C₄²×C₇	C₂×C₄	1	1	yes	characteristic
39	C₂×C₄²×C₇	D₄	1	1	yes	characteristic
40	C₂×C₄²×C₇	D₄	1	1	yes	characteristic
41	C₂×C₄²×C₇	D₄	1	1	yes	characteristic
42	C₂×C₄²×C₇	Q₈	1	1	yes	characteristic
43	C₂×C₄²×C₇	Q₈	1	1	yes	characteristic
44	C₂×C₄²×C₇	Q₈	1	1	yes	characteristic
45	C₂×C₄²×C₇	C₂³	1	1	yes	characteristic
46	C₂×C₄²×C₇	C₂³	1	1	yes	characteristic
47	C₂×C₄²×C₇	C₂³	1	1	yes	characteristic
48	C₂²×C₇⋊C₈	—	2	2	no	—
49	C₂²×C₇⋊C₈	—	2	2	no	—
50	C₇×C₄⋊C₈	C₂×C₄	1	4	yes	—
51	C₇×C₄⋊C₈	C₂×C₄	1	4	yes	—
52	C₇×C₂²⋊C₈	C₂×C₄	1	4	yes	—
53	C₇×C₂²⋊C₈	C₂×C₄	1	4	yes	—
54	C₇×#97	C₂×C₄	1	4	yes	—
55	C₄×C₇×C₈	C₂³	1	4	yes	—
56	C₂²⋊(C₇⋊C₈)	—	2	2	no	—
57	C₂²⋊(C₇⋊C₈)	—	2	2	no	—
58	C₂²⋊(C₇⋊C₈)	—	2	2	no	—
59	C₂²⋊(C₇⋊C₈)	—	2	2	no	—
60	C₄⋊(C₇⋊C₈)	—	2	2	no	—
61	C₄⋊(C₇⋊C₈)	—	2	2	no	—
62	C₄⋊(C₇⋊C₈)	C₂³	1	4	yes	—
63	C₄⋊(C₇⋊C₈)	C₂³	1	4	yes	—
64	#6377	—	2	2	no	—
65	#6377	—	2	2	no	—
66	C₄×C₇⋊C₈	—	2	2	no	—
67	C₄×C₇⋊C₈	—	2	2	no	—
68	#2920	D₇	1	1	yes	characteristic
69	C₂×#432	—	7	1	no	—
70	C₂×#431	—	7	1	no	—
71	#1573	—	7	4	no	—
72	#1572	—	7	4	no	—
73	#1565	—	7	4	no	—
74	C₂⁴×C₇	C₄⋊C₄	1	1	yes	characteristic
75	C₂²×C₄×C₇	—	2	4	no	—
76	C₂²×C₄×C₇	—	4	2	no	—
77	C₂²×C₄×C₇	—	4	2	no	—
78	C₂²×C₄×C₇	C₄⋊C₄	1	1	yes	characteristic
79	C₂²×C₄×C₇	C₄⋊C₄	1	1	yes	characteristic
80	C₂²×C₄×C₇	C₄⋊C₄	1	1	yes	characteristic
81	C₂²×C₄×C₇	C₄⋊C₄	1	1	yes	characteristic
82	C₂²×C₄×C₇	C₄⋊C₄	1	1	yes	characteristic
83	C₂²×C₄×C₇	C₄⋊C₄	1	1	yes	characteristic
84	C₂²×C₄×C₇	C₄⋊C₄	1	1	yes	characteristic
85	C₂²×C₄×C₇	C₄⋊C₄	1	1	yes	characteristic
86	C₂²×C₄×C₇	C₄⋊C₄	1	1	yes	characteristic
87	C₂²×C₄×C₇	C₂²×C₄	1	1	yes	characteristic
88	C₂²×C₄×C₇	C₂²×C₄	1	1	yes	characteristic
89	C₂²×C₄×C₇	C₂²×C₄	1	1	yes	characteristic
90	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
91	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
92	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
93	C₂²×C₄×C₇	C₂×Q₈	1	1	yes	characteristic
94	C₂²×C₄×C₇	C₂×Q₈	1	1	yes	characteristic
95	C₂²×C₄×C₇	C₂×Q₈	1	1	yes	characteristic
96	C₇×M₁₆	—	2	8	no	—
97	C₂×C₇×C₈	C₂²×C₄	1	8	yes	—
98	C₄²×C₇	C₄⋊C₄	1	1	yes	characteristic
99	C₄²×C₇	C₄⋊C₄	1	1	yes	characteristic
100	C₄²×C₇	C₄⋊C₄	1	1	yes	characteristic
101	C₄²×C₇	C₄⋊C₄	1	1	yes	characteristic
102	C₄²×C₇	C₄⋊C₄	1	1	yes	characteristic
103	C₄²×C₇	C₄⋊C₄	1	1	yes	characteristic
104	C₄²×C₇	C₂²×C₄	1	1	yes	characteristic
105	C₄²×C₇	C₂²×C₄	1	1	yes	characteristic
106	C₄²×C₇	C₂²×C₄	1	1	yes	characteristic
107	C₄²×C₇	C₂×D₄	1	1	yes	characteristic
108	C₄²×C₇	C₂×D₄	1	1	yes	characteristic
109	C₄²×C₇	C₂×D₄	1	1	yes	characteristic
110	C₄²×C₇	C₂×Q₈	1	1	yes	characteristic
111	C₄²×C₇	C₂×Q₈	1	1	yes	characteristic
112	C₄²×C₇	C₂×Q₈	1	1	yes	characteristic
113	C₄²×C₇	C₂⁴	1	1	yes	characteristic
114	C₂×C₇⋊C₈	—	2	4	no	—
115	C₂×C₇⋊C₈	—	2	4	no	—
116	C₂×C₇⋊C₈	—	4	4	no	—
117	C₂×C₇⋊C₈	—	4	4	no	—
118	C₂²×C₄²	C₂×D₇	1	1	yes	characteristic
119	#432	—	7	2	no	—
120	#432	—	7	2	no	—
121	#432	—	14	1	no	—
122	#432	—	14	1	no	—
123	#432	Dic₇	1	2	yes	—
124	#432	Dic₇	1	2	yes	—
125	#431	—	7	2	no	—
126	#431	—	7	2	no	—
127	#431	—	14	1	no	—
128	#431	—	14	1	no	—
129	#431	C₂×D₇	1	2	yes	—
130	#431	C₂×D₇	1	2	yes	—
131	#423	Dic₇	1	2	yes	—
132	C₂×C₄⋊C₈	—	7	2	no	—
133	C₂×C₄⋊C₈	—	7	2	no	—
134	#407	Dic₇	1	2	yes	—
135	C₂×C₂²⋊C₈	—	14	1	no	—
136	C₂×C₂²⋊C₈	—	14	1	no	—
137	C₄×M₁₆	C₂×D₇	1	2	yes	—
138	C₂×#97	—	14	1	no	—
139	C₂×C₄×C₈	—	14	1	no	—
140	#335	—	7	8	no	—
141	#334	—	7	8	no	—
142	C₂³×C₇	—	2	2	no	—
143	C₂³×C₇	—	2	2	no	—
144	C₂³×C₇	—	4	1	no	—
145	C₂³×C₇	C₂×C₄⋊C₄	1	1	yes	characteristic
146	C₂³×C₇	C₂×C₄⋊C₄	1	1	yes	characteristic
147	C₂³×C₇	C₂×C₄⋊C₄	1	1	yes	characteristic
148	C₂×C₄×C₇	—	2	2	no	—
149	C₂×C₄×C₇	—	2	2	no	—
150	C₂×C₄×C₇	—	2	2	no	—
151	C₂×C₄×C₇	—	2	2	no	—
152	C₂×C₄×C₇	—	2	2	no	—
153	C₂×C₄×C₇	—	2	2	no	—
154	C₂×C₄×C₇	—	2	2	no	—
155	C₂×C₄×C₇	—	2	2	no	—
156	C₂×C₄×C₇	—	2	2	no	—
157	C₂×C₄×C₇	—	2	2	no	—
158	C₂×C₄×C₇	—	4	1	no	—
159	C₂×C₄×C₇	—	4	1	no	—
160	C₂×C₄×C₇	—	4	1	no	—
161	C₂×C₄×C₇	—	4	1	no	—
162	C₂×C₄×C₇	—	4	1	no	—
163	C₂×C₄×C₇	—	4	1	no	—
164	C₂×C₄×C₇	—	4	1	no	—
165	C₂×C₄×C₇	C₂×C₄⋊C₄	1	1	yes	characteristic
166	C₂×C₄×C₇	C₂×C₄⋊C₄	1	1	yes	characteristic
167	C₂×C₄×C₇	C₂×C₄⋊C₄	1	1	yes	characteristic
168	C₂×C₄×C₇	C₂×C₄⋊C₄	1	1	yes	characteristic
169	C₂×C₄×C₇	C₂×C₄⋊C₄	1	1	yes	characteristic
170	C₂×C₄×C₇	C₂×C₄⋊C₄	1	1	yes	characteristic
171	C₂×C₄×C₇	C₂×C₄⋊C₄	1	1	yes	characteristic
172	C₂×C₄×C₇	C₂×C₄⋊C₄	1	1	yes	characteristic
173	C₂×C₄×C₇	C₂×C₄⋊C₄	1	1	yes	characteristic
174	C₂×C₄×C₇	#131	1	4	yes	—
175	C₂×C₄×C₇	C₂³×C₄	1	1	yes	characteristic
176	C₂×C₄×C₇	C₂²×D₄	1	1	yes	characteristic
177	C₂×C₄×C₇	C₂²×Q₈	1	1	yes	characteristic
178	C₇×C₈	—	2	8	no	—
179	C₇⋊C₈	—	4	4	no	—
180	C₇⋊C₈	—	4	4	no	—
181	C₂³×C₄	Dic₁₄	1	1	yes	characteristic
182	C₂³×C₄	D₂₈	1	1	yes	characteristic
183	C₂³×C₄	C₂×Dic₇	1	1	yes	characteristic
184	C₂×M₁₆	C₂×Dic₇	1	4	yes	—
185	C₂²×C₈	—	14	2	no	—
186	C₂²×C₈	—	14	2	no	—
187	C₂×C₄²	Dic₁₄	1	1	yes	characteristic
188	C₂×C₄²	Dic₁₄	1	1	yes	characteristic
189	C₂×C₄²	Dic₁₄	1	1	yes	characteristic
190	C₂×C₄²	D₂₈	1	1	yes	characteristic
191	C₂×C₄²	D₂₈	1	1	yes	characteristic
192	C₂×C₄²	D₂₈	1	1	yes	characteristic
193	C₂×C₄²	C₂×Dic₇	1	1	yes	characteristic
194	C₂×C₄²	C₂×Dic₇	1	1	yes	characteristic
195	C₂×C₄²	C₂×Dic₇	1	1	yes	characteristic
196	C₂×C₄²	C₂²×D₇	1	1	yes	characteristic
197	C₂×C₄²	C₂²×D₇	1	1	yes	characteristic
198	C₂×C₄²	C₂²×D₇	1	1	yes	characteristic
199	C₄⋊C₈	—	7	4	no	—
200	C₄⋊C₈	—	7	4	no	—
201	C₄⋊C₈	—	14	2	no	—
202	C₄⋊C₈	—	14	2	no	—
203	C₄⋊C₈	C₂×Dic₇	1	4	yes	—
204	C₄⋊C₈	C₂×Dic₇	1	4	yes	—
205	C₂²⋊C₈	—	14	2	no	—
206	C₂²⋊C₈	—	14	2	no	—
207	C₂²⋊C₈	—	14	2	no	—
208	C₂²⋊C₈	—	14	2	no	—
209	C₂²⋊C₈	C₂×Dic₇	1	4	yes	—
210	C₂²⋊C₈	C₂×Dic₇	1	4	yes	—
211	#97	—	14	2	no	—
212	#97	—	14	2	no	—
213	#97	C₂×Dic₇	1	4	yes	—
214	C₄×C₈	—	14	2	no	—
215	C₄×C₈	—	14	2	no	—
216	C₄×C₈	C₂²×D₇	1	4	yes	—
217	C₂²×C₇	—	2	1	no	—
218	C₂²×C₇	—	2	1	no	—
219	C₂²×C₇	—	2	1	no	—
220	C₂²×C₇	—	2	1	no	—
221	C₂²×C₇	—	2	1	no	—
222	C₂²×C₇	—	2	1	no	—
223	C₂²×C₇	—	2	1	no	—
224	C₂²×C₇	—	4	4	no	—
225	C₂²×C₇	#424	1	2	yes	—
226	C₂²×C₇	#430	1	2	yes	—
227	C₂²×C₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
228	C₄×C₇	—	2	1	no	—
229	C₄×C₇	—	2	1	no	—
230	C₄×C₇	—	2	1	no	—
231	C₄×C₇	—	2	1	no	—
232	C₄×C₇	—	2	1	no	—
233	C₄×C₇	—	2	1	no	—
234	C₄×C₇	—	2	1	no	—
235	C₄×C₇	—	2	1	no	—
236	C₄×C₇	—	2	1	no	—
237	C₄×C₇	#424	1	2	yes	—
238	C₄×C₇	C₂×#131	1	2	yes	—
239	C₄×C₇	#575	1	2	yes	—
240	C₂⁴	C₄⋊Dic₇	1	1	yes	characteristic
241	C₂²×C₄	—	2	4	no	—
242	C₂²×C₄	—	4	2	no	—
243	C₂²×C₄	—	4	2	no	—
244	C₂²×C₄	C₄⋊Dic₇	1	1	yes	characteristic
245	C₂²×C₄	C₄⋊Dic₇	1	1	yes	characteristic
246	C₂²×C₄	C₄⋊Dic₇	1	1	yes	characteristic
247	C₂²×C₄	C₄⋊Dic₇	1	1	yes	characteristic
248	C₂²×C₄	C₄⋊Dic₇	1	1	yes	characteristic
249	C₂²×C₄	C₄⋊Dic₇	1	1	yes	characteristic
250	C₂²×C₄	C₄⋊Dic₇	1	1	yes	characteristic
251	C₂²×C₄	C₄⋊Dic₇	1	1	yes	characteristic
252	C₂²×C₄	C₄⋊Dic₇	1	1	yes	characteristic
253	C₂²×C₄	C₂×Dic₁₄	1	1	yes	characteristic
254	C₂²×C₄	C₂×Dic₁₄	1	1	yes	characteristic
255	C₂²×C₄	C₂×Dic₁₄	1	1	yes	characteristic
256	C₂²×C₄	C₂×D₂₈	1	1	yes	characteristic
257	C₂²×C₄	C₂×D₂₈	1	1	yes	characteristic
258	C₂²×C₄	C₂×D₂₈	1	1	yes	characteristic
259	C₂²×C₄	C₂²×Dic₇	1	1	yes	characteristic
260	C₂²×C₄	C₂²×Dic₇	1	1	yes	characteristic
261	C₂²×C₄	C₂²×Dic₇	1	1	yes	characteristic
262	M₁₆	—	2	8	no	—
263	C₂×C₈	—	14	4	no	—
264	C₂×C₈	—	14	4	no	—
265	C₂×C₈	—	28	4	no	—
266	C₂×C₈	—	28	4	no	—
267	C₂×C₈	C₂²×Dic₇	1	8	yes	—
268	C₄²	C₄⋊Dic₇	1	1	yes	characteristic
269	C₄²	C₄⋊Dic₇	1	1	yes	characteristic
270	C₄²	C₄⋊Dic₇	1	1	yes	characteristic
271	C₄²	C₄⋊Dic₇	1	1	yes	characteristic
272	C₄²	C₄⋊Dic₇	1	1	yes	characteristic
273	C₄²	C₄⋊Dic₇	1	1	yes	characteristic
274	C₄²	C₂×Dic₁₄	1	1	yes	characteristic
275	C₄²	C₂×Dic₁₄	1	1	yes	characteristic
276	C₄²	C₂×Dic₁₄	1	1	yes	characteristic
277	C₄²	C₂×D₂₈	1	1	yes	characteristic
278	C₄²	C₂×D₂₈	1	1	yes	characteristic
279	C₄²	C₂×D₂₈	1	1	yes	characteristic
280	C₄²	C₂²×Dic₇	1	1	yes	characteristic
281	C₄²	C₂²×Dic₇	1	1	yes	characteristic
282	C₄²	C₂²×Dic₇	1	1	yes	characteristic
283	C₄²	C₂³×D₇	1	1	yes	characteristic
284	C₂×C₇	—	2	2	no	—
285	C₂×C₇	—	2	2	no	—
286	C₂×C₇	—	4	1	no	—
287	C₂×C₇	C₂×#424	1	1	yes	characteristic
288	C₂×C₇	#2911	1	1	yes	characteristic
289	C₂×C₇	C₂×#430	1	1	yes	characteristic
290	C₂³	—	2	2	no	—
291	C₂³	—	2	2	no	—
292	C₂³	—	4	1	no	—
293	C₂³	C₂×C₄⋊Dic₇	1	1	yes	characteristic
294	C₂³	C₂×C₄⋊Dic₇	1	1	yes	characteristic
295	C₂³	C₂×C₄⋊Dic₇	1	1	yes	characteristic
296	C₂×C₄	—	2	2	no	—
297	C₂×C₄	—	2	2	no	—
298	C₂×C₄	—	2	2	no	—
299	C₂×C₄	—	2	2	no	—
300	C₂×C₄	—	2	2	no	—
301	C₂×C₄	—	2	2	no	—
302	C₂×C₄	—	2	2	no	—
303	C₂×C₄	—	2	2	no	—
304	C₂×C₄	—	2	2	no	—
305	C₂×C₄	—	2	2	no	—
306	C₂×C₄	—	4	1	no	—
307	C₂×C₄	—	4	1	no	—
308	C₂×C₄	—	4	1	no	—
309	C₂×C₄	—	4	1	no	—
310	C₂×C₄	—	4	1	no	—
311	C₂×C₄	—	4	1	no	—
312	C₂×C₄	—	4	1	no	—
313	C₂×C₄	C₂×C₄⋊Dic₇	1	1	yes	characteristic
314	C₂×C₄	C₂×C₄⋊Dic₇	1	1	yes	characteristic
315	C₂×C₄	C₂×C₄⋊Dic₇	1	1	yes	characteristic
316	C₂×C₄	C₂×C₄⋊Dic₇	1	1	yes	characteristic
317	C₂×C₄	C₂×C₄⋊Dic₇	1	1	yes	characteristic
318	C₂×C₄	C₂×C₄⋊Dic₇	1	1	yes	characteristic
319	C₂×C₄	C₂×C₄⋊Dic₇	1	1	yes	characteristic
320	C₂×C₄	C₂×C₄⋊Dic₇	1	1	yes	characteristic
321	C₂×C₄	C₂×C₄⋊Dic₇	1	1	yes	characteristic
322	C₂×C₄	#6511	1	4	yes	—
323	C₂×C₄	C₂²×Dic₁₄	1	1	yes	characteristic
324	C₂×C₄	C₂²×D₂₈	1	1	yes	characteristic
325	C₂×C₄	C₂³×Dic₇	1	1	yes	characteristic
326	C₈	—	2	8	no	—
327	C₈	—	28	4	no	—
328	C₈	—	28	4	no	—
329	C₇	#17810	1	1	yes	characteristic, complemented, Sylow
330	C₂²	—	2	1	no	—
331	C₂²	—	2	1	no	—
332	C₂²	—	2	1	no	—
333	C₂²	—	2	1	no	—
334	C₂²	—	2	1	no	—
335	C₂²	—	2	1	no	—
336	C₂²	—	2	1	no	—
337	C₂²	—	4	4	no	—
338	C₂²	#89771	1	2	yes	—
339	C₂²	#89897	1	2	yes	—
340	C₂²	C₂²×C₄⋊Dic₇	1	1	yes	characteristic
341	C₄	—	2	1	no	—
342	C₄	—	2	1	no	—
343	C₄	—	2	1	no	—
344	C₄	—	2	1	no	—
345	C₄	—	2	1	no	—
346	C₄	—	2	1	no	—
347	C₄	—	2	1	no	—
348	C₄	—	2	1	no	—
349	C₄	—	2	1	no	—
350	C₄	#89771	1	2	yes	—
351	C₄	#90439	1	2	yes	—
352	C₄	C₂×#6511	1	2	yes	—
353	C₂	—	2	2	no	—
354	C₂	—	2	2	no	—
355	C₂	—	4	1	no	—
356	C₂	C₂×#89771	1	1	yes	characteristic
357	C₂	#11741197	1	1	yes	characteristic
358	C₂	C₂×#89897	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂²×D₂₈
commutator	C₂×C₄×C₇	C₂³×C₄
Frattini	C₄²	C₂³×D₇
Fitting	C₇×#2920	C₂
socle	C₂²×C₇	C₂²×C₄⋊C₄
prime radical	C₂⁴×C₇	C₄⋊C₄
cosocle	C₄²×C₇	C₂⁴
hypercenter	#2920	D₇
hypocenter	C₇	#17810
LCS	G ▷ C₂×C₄×C₇ ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ #2920

Complements

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

N	H ≅ G/N	type
4. #11728506	C₂	semidirect
5. #11728505	C₂	semidirect
6. #11727604	C₂	semidirect
29. #89253	C₂²	semidirect
30. #89252	C₂²	semidirect
329. C₇	#17810	semidirect

Maximal subgroups

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

subgroup	index
C₇×#2920	2
C₂×#89698	2
C₂×#89697	2
#11728506	2
#11728505	2
#11727604	2
#17810	7