Subgroups — #13244007

Order 1792.

698 conjugacy classes of subgroups.

The subgroup lattice has 370 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×#89704	C₂	1	1	yes	maximal, characteristic
3	#11739494	C₂	1	1	yes	maximal, characteristic
4	#11725554	C₂	1	4	yes	maximal, complemented
5	C₂²×C₄²×C₇	C₂²	1	1	yes	characteristic
6	C₂²×C₄×Dic₇	—	2	1	no	—
7	C₇×#432	—	2	1	no	—
8	C₇×#432	C₂²	1	2	yes	—
9	C₇×#431	—	2	1	no	—
10	C₇×#431	C₄	1	2	yes	complemented
11	C₇×#423	C₄	1	2	yes	complemented
12	C₇×#407	—	2	1	no	—
13	C₄×C₇×M₁₆	—	2	1	no	—
14	C₂×C₇⋊#95	—	2	1	no	—
15	C₂×C₇⋊#95	C₄	1	2	yes	—
16	#89979	—	2	1	no	—
17	#89704	—	2	1	no	—
18	#89704	—	2	1	no	—
19	#89704	C₄	1	2	yes	—
20	#89704	C₂²	1	2	yes	—
21	#89698	C₄	1	2	yes	complemented
22	#89698	C₄	1	2	yes	complemented
23	#89697	—	2	1	no	—
24	#89697	—	2	1	no	—
25	#89696	—	2	1	no	—
26	C₄×#1136	C₂²	1	2	yes	—
27	#8072	—	7	1	no	maximal, Sylow
28	C₂³×C₄×C₇	C₂×C₄	1	1	yes	characteristic
29	C₂³×C₄×C₇	D₄	1	1	yes	characteristic
30	C₂³×C₄×C₇	D₄	1	1	yes	characteristic
31	C₂³×Dic₇	—	2	2	no	—
32	C₂×C₇×M₁₆	—	2	2	no	—
33	C₂×C₄²×C₇	C₂×C₄	1	1	yes	characteristic
34	C₂×C₄²×C₇	C₂×C₄	1	1	yes	characteristic
35	C₂×C₄²×C₇	C₂×C₄	1	1	yes	characteristic
36	C₂×C₄²×C₇	C₂×C₄	1	1	yes	characteristic
37	C₂×C₄²×C₇	C₂×C₄	1	1	yes	characteristic
38	C₂×C₄²×C₇	D₄	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	C₂³	1	1	yes	characteristic
45	C₂×C₄×Dic₇	—	2	8	no	—
46	C₂×C₄×Dic₇	—	2	2	no	—
47	C₂×C₄×Dic₇	—	2	2	no	—
48	C₂×#1136	—	2	2	no	—
49	C₇×C₄⋊C₈	—	2	2	no	—
50	C₇×C₄⋊C₈	C₂×C₄	1	4	yes	complemented
51	C₇×C₂²⋊C₈	—	2	2	no	—
52	C₇×C₂²⋊C₈	—	2	2	no	—
53	C₇×#97	—	2	2	no	—
54	C₄×C₇×C₈	—	2	2	no	—
55	C₇⋊#95	—	2	2	no	—
56	C₇⋊#95	—	2	2	no	—
57	C₇⋊#95	—	2	2	no	—
58	C₇⋊#95	C₂×C₄	1	4	yes	—
59	C₂²⋊(C₇⋊C₈)	—	2	2	no	—
60	C₂²⋊(C₇⋊C₈)	—	2	2	no	—
61	C₄⋊(C₇⋊C₈)	—	2	2	no	—
62	C₄⋊(C₇⋊C₈)	—	2	2	no	—
63	#6377	C₂×C₄	1	4	yes	complemented
64	C₄×C₇⋊C₈	—	2	2	no	—
65	#2920	—	7	1	no	—
66	#2920	D₇	1	1	yes	characteristic
67	C₂×#376	—	7	1	no	—
68	#1300	—	7	4	no	—
69	C₂⁴×C₇	C₂²⋊C₄	1	1	yes	characteristic
70	C₂²×C₄×C₇	—	2	4	no	—
71	C₂²×C₄×C₇	—	4	2	no	—
72	C₂²×C₄×C₇	—	4	2	no	—
73	C₂²×C₄×C₇	C₄²	1	1	yes	characteristic
74	C₂²×C₄×C₇	C₄²	1	1	yes	characteristic
75	C₂²×C₄×C₇	C₂²⋊C₄	1	1	yes	characteristic
76	C₂²×C₄×C₇	C₂²⋊C₄	1	1	yes	characteristic
77	C₂²×C₄×C₇	C₂²⋊C₄	1	1	yes	characteristic
78	C₂²×C₄×C₇	C₂²⋊C₄	1	1	yes	characteristic
79	C₂²×C₄×C₇	C₂²⋊C₄	1	1	yes	characteristic
80	C₂²×C₄×C₇	C₂²⋊C₄	1	1	yes	characteristic
81	C₂²×C₄×C₇	C₂²⋊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	C₄⋊C₄	1	1	yes	characteristic
88	C₂²×C₄×C₇	C₂²×C₄	1	1	yes	characteristic
89	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
90	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
91	C₂²×Dic₇	—	2	8	no	—
92	C₂²×Dic₇	—	2	4	no	—
93	C₂²×Dic₇	—	4	4	no	—
94	C₂²×Dic₇	—	4	4	no	—
95	C₇×M₁₆	—	4	4	no	—
96	C₂×C₇×C₈	—	2	4	no	—
97	C₄²×C₇	C₄²	1	1	yes	characteristic
98	C₄²×C₇	C₄²	1	1	yes	characteristic
99	C₄²×C₇	C₂²⋊C₄	1	1	yes	characteristic
100	C₄²×C₇	C₂²⋊C₄	1	1	yes	characteristic
101	C₄²×C₇	C₂²⋊C₄	1	1	yes	characteristic
102	C₄²×C₇	C₂²⋊C₄	1	1	yes	characteristic
103	C₄²×C₇	C₄⋊C₄	1	1	yes	characteristic
104	C₄²×C₇	C₄⋊C₄	1	1	yes	characteristic
105	C₄²×C₇	C₄⋊C₄	1	1	yes	characteristic
106	C₄²×C₇	C₄⋊C₄	1	1	yes	characteristic
107	C₄²×C₇	C₄⋊C₄	1	1	yes	characteristic
108	C₄²×C₇	C₄⋊C₄	1	1	yes	characteristic
109	C₄²×C₇	C₂²×C₄	1	1	yes	characteristic
110	C₄²×C₇	C₂²×C₄	1	1	yes	characteristic
111	C₄²×C₇	C₂×D₄	1	1	yes	characteristic
112	C₄²×C₇	C₂×Q₈	1	1	yes	characteristic
113	C₄×Dic₇	—	2	8	no	—
114	C₄×Dic₇	—	2	8	no	—
115	#1136	—	4	4	no	—
116	C₂×C₇⋊C₈	—	2	4	no	—
117	C₂²×C₄²	—	14	1	no	—
118	C₂²×C₄²	C₂×D₇	1	1	yes	characteristic
119	#432	—	2	1	no	—
120	#432	—	7	2	no	—
121	#432	—	7	2	no	—
122	#432	C₂×D₇	1	2	yes	—
123	#431	—	2	1	no	—
124	#431	—	14	1	no	—
125	#431	—	14	1	no	—
126	#431	Dic₇	1	2	yes	complemented
127	#423	—	14	1	no	—
128	#423	Dic₇	1	2	yes	complemented
129	#407	—	2	1	no	—
130	#407	—	14	1	no	—
131	C₄×M₁₆	—	2	1	no	—
132	C₄×M₁₆	—	7	2	no	—
133	#376	—	7	2	no	—
134	#376	—	7	2	no	—
135	#376	—	14	1	no	—
136	#376	—	14	1	no	—
137	C₂×#95	—	7	2	no	—
138	C₂×#95	—	14	1	no	—
139	C₂³×C₇	—	2	2	no	—
140	C₂³×C₇	—	2	2	no	—
141	C₂³×C₇	—	2	2	no	—
142	C₂³×C₇	#95	1	1	yes	characteristic
143	C₂³×C₇	#95	1	1	yes	characteristic
144	C₂³×C₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
145	C₂×C₄×C₇	—	2	2	no	—
146	C₂×C₄×C₇	—	2	2	no	—
147	C₂×C₄×C₇	—	2	2	no	—
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	—	4	1	no	—
155	C₂×C₄×C₇	—	4	1	no	—
156	C₂×C₄×C₇	—	4	1	no	—
157	C₂×C₄×C₇	—	4	1	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	#95	1	1	yes	characteristic
163	C₂×C₄×C₇	#95	1	1	yes	characteristic
164	C₂×C₄×C₇	#95	1	1	yes	characteristic
165	C₂×C₄×C₇	#95	1	1	yes	characteristic
166	C₂×C₄×C₇	#95	1	1	yes	characteristic
167	C₂×C₄×C₇	#95	1	1	yes	characteristic
168	C₂×C₄×C₇	#104	1	4	yes	—
169	C₂×C₄×C₇	C₂×C₄²	1	1	yes	characteristic
170	C₂×C₄×C₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
171	C₂×C₄×C₇	C₂×C₂²⋊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	C₂×C₄⋊C₄	1	1	yes	characteristic
175	C₂×Dic₇	—	2	8	no	—
176	C₂×Dic₇	—	4	16	no	—
177	C₂×Dic₇	—	4	4	no	—
178	C₂×Dic₇	—	4	4	no	—
179	C₇×C₈	—	4	4	no	—
180	C₇⋊C₈	—	4	4	no	—
181	C₂³×C₄	—	14	2	no	—
182	C₂³×C₄	C₄×D₇	1	1	yes	characteristic
183	C₂³×C₄	D₂₈	1	1	yes	characteristic
184	C₂³×C₄	C₂²⋊D₇	1	1	yes	characteristic
185	C₂×M₁₆	—	2	2	no	—
186	C₂×M₁₆	—	14	2	no	—
187	C₂×C₄²	—	14	8	no	—
188	C₂×C₄²	—	14	2	no	—
189	C₂×C₄²	—	14	2	no	—
190	C₂×C₄²	Dic₁₄	1	1	yes	characteristic
191	C₂×C₄²	Dic₁₄	1	1	yes	characteristic
192	C₂×C₄²	C₄×D₇	1	1	yes	characteristic
193	C₂×C₄²	C₄×D₇	1	1	yes	characteristic
194	C₂×C₄²	C₄×D₇	1	1	yes	characteristic
195	C₂×C₄²	D₂₈	1	1	yes	characteristic
196	C₂×C₄²	C₂×Dic₇	1	1	yes	characteristic
197	C₂×C₄²	C₂×Dic₇	1	1	yes	characteristic
198	C₂×C₄²	C₂²⋊D₇	1	1	yes	characteristic
199	C₂×C₄²	C₂²⋊D₇	1	1	yes	characteristic
200	C₂×C₄²	C₂²⋊D₇	1	1	yes	characteristic
201	C₂×C₄²	C₂²×D₇	1	1	yes	characteristic
202	C₄⋊C₈	—	2	2	no	—
203	C₄⋊C₈	—	14	2	no	—
204	C₄⋊C₈	—	14	2	no	—
205	C₄⋊C₈	C₂×Dic₇	1	4	yes	complemented
206	C₂²⋊C₈	—	2	2	no	—
207	C₂²⋊C₈	—	2	2	no	—
208	C₂²⋊C₈	—	14	2	no	—
209	C₂²⋊C₈	—	14	2	no	—
210	#97	—	2	2	no	—
211	#97	—	7	4	no	—
212	C₄×C₈	—	2	2	no	—
213	C₄×C₈	—	14	2	no	—
214	#95	—	7	4	no	—
215	#95	—	14	2	no	—
216	#95	—	14	2	no	—
217	#95	—	14	2	no	—
218	C₂²×C₇	—	2	8	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	—	2	1	no	—
225	C₂²×C₇	#339	1	2	yes	—
226	C₂²×C₇	#344	1	2	yes	—
227	C₂²×C₇	C₂×#95	1	1	yes	characteristic
228	C₂²×C₇	#421	1	2	yes	—
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	—	2	1	no	—
238	C₄×C₇	—	2	1	no	—
239	C₄×C₇	#339	1	2	yes	—
240	C₄×C₇	C₂×#104	1	2	yes	—
241	Dic₇	—	4	8	no	—
242	C₂⁴	(C₂×D₇)⋊C₄	1	1	yes	characteristic
243	C₂²×C₄	—	2	4	no	—
244	C₂²×C₄	—	4	2	no	—
245	C₂²×C₄	—	4	2	no	—
246	C₂²×C₄	—	14	8	no	—
247	C₂²×C₄	—	14	4	no	—
248	C₂²×C₄	—	28	4	no	—
249	C₂²×C₄	—	28	4	no	—
250	C₂²×C₄	C₄×Dic₇	1	1	yes	characteristic
251	C₂²×C₄	C₄×Dic₇	1	1	yes	characteristic
252	C₂²×C₄	Dic₇⋊C₄	1	1	yes	characteristic
253	C₂²×C₄	Dic₇⋊C₄	1	1	yes	characteristic
254	C₂²×C₄	Dic₇⋊C₄	1	1	yes	characteristic
255	C₂²×C₄	Dic₇⋊C₄	1	1	yes	characteristic
256	C₂²×C₄	C₄⋊Dic₇	1	1	yes	characteristic
257	C₂²×C₄	C₄⋊Dic₇	1	1	yes	characteristic
258	C₂²×C₄	(C₂×D₇)⋊C₄	1	1	yes	characteristic
259	C₂²×C₄	(C₂×D₇)⋊C₄	1	1	yes	characteristic
260	C₂²×C₄	(C₂×D₇)⋊C₄	1	1	yes	characteristic
261	C₂²×C₄	(C₂×D₇)⋊C₄	1	1	yes	characteristic
262	C₂²×C₄	(C₂×D₇)⋊C₄	1	1	yes	characteristic
263	C₂²×C₄	C₂²⋊Dic₇	1	1	yes	characteristic
264	C₂²×C₄	C₂²⋊Dic₇	1	1	yes	characteristic
265	C₂²×C₄	C₂×C₄×D₇	1	1	yes	characteristic
266	C₂²×C₄	C₂×D₂₈	1	1	yes	characteristic
267	C₂²×C₄	C₂×C₂²⋊D₇	1	1	yes	characteristic
268	M₁₆	—	4	4	no	—
269	M₁₆	—	28	4	no	—
270	C₂×C₈	—	2	4	no	—
271	C₂×C₈	—	14	4	no	—
272	C₄²	—	14	8	no	—
273	C₄²	—	14	8	no	—
274	C₄²	C₄×Dic₇	1	1	yes	characteristic
275	C₄²	C₄×Dic₇	1	1	yes	characteristic
276	C₄²	Dic₇⋊C₄	1	1	yes	characteristic
277	C₄²	Dic₇⋊C₄	1	1	yes	characteristic
278	C₄²	Dic₇⋊C₄	1	1	yes	characteristic
279	C₄²	Dic₇⋊C₄	1	1	yes	characteristic
280	C₄²	C₄⋊Dic₇	1	1	yes	characteristic
281	C₄²	C₄⋊Dic₇	1	1	yes	characteristic
282	C₄²	(C₂×D₇)⋊C₄	1	1	yes	characteristic
283	C₄²	(C₂×D₇)⋊C₄	1	1	yes	characteristic
284	C₄²	C₂²⋊Dic₇	1	1	yes	characteristic
285	C₄²	C₂²⋊Dic₇	1	1	yes	characteristic
286	C₄²	C₂×Dic₁₄	1	1	yes	characteristic
287	C₄²	C₂×C₄×D₇	1	1	yes	characteristic
288	C₄²	C₂²×Dic₇	1	1	yes	characteristic
289	C₄²	C₂×C₂²⋊D₇	1	1	yes	characteristic
290	C₂×C₇	—	2	2	no	—
291	C₂×C₇	—	2	2	no	—
292	C₂×C₇	—	2	2	no	—
293	C₂×C₇	C₂×#339	1	1	yes	characteristic
294	C₂×C₇	#1733	1	1	yes	characteristic
295	C₂×C₇	C₂×#344	1	1	yes	characteristic
296	C₂³	—	2	2	no	—
297	C₂³	—	2	2	no	—
298	C₂³	—	2	2	no	—
299	C₂³	C₇⋊#95	1	1	yes	characteristic
300	C₂³	C₇⋊#95	1	1	yes	characteristic
301	C₂³	C₂×(C₂×D₇)⋊C₄	1	1	yes	characteristic
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	—	2	2	no	—
307	C₂×C₄	—	2	2	no	—
308	C₂×C₄	—	2	2	no	—
309	C₂×C₄	—	2	2	no	—
310	C₂×C₄	—	2	2	no	—
311	C₂×C₄	—	4	1	no	—
312	C₂×C₄	—	4	1	no	—
313	C₂×C₄	—	4	1	no	—
314	C₂×C₄	—	4	1	no	—
315	C₂×C₄	—	4	1	no	—
316	C₂×C₄	—	4	1	no	—
317	C₂×C₄	—	4	1	no	—
318	C₂×C₄	—	4	1	no	—
319	C₂×C₄	—	14	8	no	—
320	C₂×C₄	—	28	16	no	—
321	C₂×C₄	—	28	4	no	—
322	C₂×C₄	—	28	4	no	—
323	C₂×C₄	C₇⋊#95	1	1	yes	characteristic
324	C₂×C₄	C₇⋊#95	1	1	yes	characteristic
325	C₂×C₄	C₇⋊#95	1	1	yes	characteristic
326	C₂×C₄	C₇⋊#95	1	1	yes	characteristic
327	C₂×C₄	C₇⋊#95	1	1	yes	characteristic
328	C₂×C₄	C₇⋊#95	1	1	yes	characteristic
329	C₂×C₄	#6411	1	4	yes	—
330	C₂×C₄	C₂×C₄×Dic₇	1	1	yes	characteristic
331	C₂×C₄	C₂×Dic₇⋊C₄	1	1	yes	characteristic
332	C₂×C₄	C₂×Dic₇⋊C₄	1	1	yes	characteristic
333	C₂×C₄	C₂×C₄⋊Dic₇	1	1	yes	characteristic
334	C₂×C₄	C₂×(C₂×D₇)⋊C₄	1	1	yes	characteristic
335	C₂×C₄	C₂×C₂²⋊Dic₇	1	1	yes	characteristic
336	C₈	—	4	4	no	—
337	C₈	—	28	4	no	—
338	C₇	#8072	1	1	yes	characteristic, complemented, Sylow
339	C₂²	—	2	8	no	—
340	C₂²	—	2	1	no	—
341	C₂²	—	2	1	no	—
342	C₂²	—	2	1	no	—
343	C₂²	—	2	1	no	—
344	C₂²	—	2	1	no	—
345	C₂²	—	2	1	no	—
346	C₂²	#89327	1	2	yes	—
347	C₂²	#89351	1	2	yes	—
348	C₂²	#89911	1	2	yes	—
349	C₂²	C₂×C₇⋊#95	1	1	yes	characteristic
350	C₄	—	2	1	no	—
351	C₄	—	2	1	no	—
352	C₄	—	2	1	no	—
353	C₄	—	2	1	no	—
354	C₄	—	2	1	no	—
355	C₄	—	2	1	no	—
356	C₄	—	2	1	no	—
357	C₄	—	2	1	no	—
358	C₄	—	2	1	no	—
359	C₄	—	2	1	no	—
360	C₄	—	28	8	no	—
361	C₄	#89327	1	2	yes	—
362	C₄	C₂×#6411	1	2	yes	—
363	C₂	—	2	2	no	—
364	C₂	—	2	2	no	—
365	C₂	—	2	2	no	—
366	C₂	C₂×#89327	1	1	yes	characteristic
367	C₂	#11729736	1	1	yes	characteristic
368	C₂	C₂×#89351	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×C₂²⋊Dic₇
commutator	C₂×C₄×C₇	C₂×C₄²
Frattini	C₂×C₄²	C₂²×D₇
Fitting	C₇×#2920	C₂
socle	C₂²×C₇	C₂×#95
prime radical	C₂⁴×C₇	C₂²⋊C₄
cosocle	C₂×C₄²×C₇	C₂³
hypercenter	#2920	D₇
hypocenter	C₇	#8072
LCS	G ▷ C₂×C₄×C₇ ▷ C₂²×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. #11725554	C₂	semidirect
10. C₇×#431	C₄	semidirect
11. C₇×#423	C₄	semidirect
21. #89698	C₄	semidirect
22. #89698	C₄	semidirect
50. C₇×C₄⋊C₈	C₂×C₄	semidirect
63. #6377	C₂×C₄	semidirect
126. #431	Dic₇	semidirect
128. #423	Dic₇	semidirect
205. C₄⋊C₈	C₂×Dic₇	semidirect
338. C₇	#8072	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×#89704	2
#11739494	2
#11725554	2
#8072	7