Subgroups — #13244201

Order 1792.

522 conjugacy classes of subgroups.

The subgroup lattice has 352 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	#11739494	C₂	1	1	yes	maximal, characteristic
3	C₂×#89698	C₂	1	1	yes	maximal, characteristic
4	C₇⋊#1308	C₂	1	4	yes	maximal, complemented
5	C₂²×C₄²×C₇	C₂²	1	1	yes	characteristic
6	C₇×#432	—	2	1	no	—
7	C₇×#432	C₂²	1	2	yes	—
8	C₇×#431	—	2	1	no	—
9	C₇×#431	C₄	1	2	yes	—
10	C₇×#423	—	2	1	no	—
11	C₇×#407	C₄	1	2	yes	—
12	C₄×C₇×M₁₆	—	2	1	no	—
13	#89979	C₄	1	2	yes	—
14	C₂×C₂²⋊(C₇⋊C₈)	C₄	1	2	yes	—
15	#89698	—	2	1	no	—
16	#89698	—	2	1	no	—
17	#89698	—	2	1	no	—
18	#89698	C₄	1	2	yes	—
19	#89698	C₂²	1	2	yes	—
20	#89698	C₂²	1	2	yes	—
21	#89697	—	2	1	no	—
22	#89697	C₄	1	2	yes	—
23	#89696	—	2	1	no	—
24	C₂×C₄⋊(C₇⋊C₈)	—	2	1	no	—
25	C₄×#1136	—	2	1	no	—
26	C₂×#6377	—	2	1	no	—
27	#8124	—	7	1	no	maximal, Sylow
28	C₂³×C₄×C₇	C₂×C₄	1	1	yes	characteristic
29	C₂³×C₄×C₇	C₂×C₄	1	1	yes	characteristic
30	C₂³×C₄×C₇	C₂×C₄	1	1	yes	characteristic
31	C₂×C₇×M₁₆	—	2	2	no	—
32	C₂×C₄²×C₇	C₂×C₄	1	1	yes	characteristic
33	C₂×C₄²×C₇	C₂×C₄	1	1	yes	characteristic
34	C₂×C₄²×C₇	C₂×C₄	1	1	yes	characteristic
35	C₂×C₄²×C₇	D₄	1	1	yes	characteristic
36	C₂×C₄²×C₇	D₄	1	1	yes	characteristic
37	C₂×C₄²×C₇	D₄	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	Q₈	1	1	yes	characteristic
42	C₂×C₄²×C₇	Q₈	1	1	yes	characteristic
43	C₂×C₄²×C₇	C₂³	1	1	yes	characteristic
44	C₂×#1136	—	2	2	no	—
45	C₂²×C₇⋊C₈	—	2	2	no	—
46	C₇×C₄⋊C₈	—	2	2	no	—
47	C₇×C₄⋊C₈	—	2	2	no	—
48	C₇×C₂²⋊C₈	—	2	2	no	—
49	C₇×C₂²⋊C₈	C₂×C₄	1	4	yes	—
50	C₇×#97	—	2	2	no	—
51	C₄×C₇×C₈	—	2	2	no	—
52	C₂²⋊(C₇⋊C₈)	—	2	2	no	—
53	C₂²⋊(C₇⋊C₈)	—	2	2	no	—
54	C₂²⋊(C₇⋊C₈)	C₂×C₄	1	4	yes	—
55	C₂²⋊(C₇⋊C₈)	C₂×C₄	1	4	yes	—
56	C₄⋊(C₇⋊C₈)	—	2	2	no	—
57	C₄⋊(C₇⋊C₈)	—	2	2	no	—
58	C₄⋊(C₇⋊C₈)	—	2	2	no	—
59	C₄⋊(C₇⋊C₈)	—	2	2	no	—
60	#6377	—	2	2	no	—
61	#6377	—	2	2	no	—
62	#6377	—	2	2	no	—
63	C₄×C₇⋊C₈	—	2	2	no	—
64	#2920	—	7	1	no	—
65	#2920	D₇	1	1	yes	characteristic
66	C₂×#432	—	7	1	no	—
67	#1308	—	7	4	no	—
68	C₂⁴×C₇	C₄²	1	1	yes	characteristic
69	C₂²×C₄×C₇	—	4	2	no	—
70	C₂²×C₄×C₇	—	4	2	no	—
71	C₂²×C₄×C₇	—	4	2	no	—
72	C₂²×C₄×C₇	C₄²	1	1	yes	characteristic
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	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	C₇×M₁₆	—	4	4	no	—
91	C₂×C₇×C₈	—	2	4	no	—
92	C₄²×C₇	C₂²⋊C₄	1	1	yes	characteristic
93	C₄²×C₇	C₂²⋊C₄	1	1	yes	characteristic
94	C₄²×C₇	C₂²⋊C₄	1	1	yes	characteristic
95	C₄²×C₇	C₂²⋊C₄	1	1	yes	characteristic
96	C₄²×C₇	C₂²⋊C₄	1	1	yes	characteristic
97	C₄²×C₇	C₂²⋊C₄	1	1	yes	characteristic
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	C₂×D₄	1	1	yes	characteristic
105	C₄²×C₇	C₂×D₄	1	1	yes	characteristic
106	C₄²×C₇	C₂×D₄	1	1	yes	characteristic
107	C₄²×C₇	C₂×Q₈	1	1	yes	characteristic
108	#1136	—	4	4	no	—
109	C₂×C₇⋊C₈	—	2	4	no	—
110	C₂×C₇⋊C₈	—	2	4	no	—
111	C₂×C₇⋊C₈	—	4	4	no	—
112	C₂²×C₄²	C₂×D₇	1	1	yes	characteristic
113	#432	—	2	1	no	—
114	#432	—	7	2	no	—
115	#432	—	7	2	no	—
116	#432	—	7	2	no	—
117	#432	—	14	1	no	—
118	#432	—	14	1	no	—
119	#432	—	14	1	no	—
120	#432	C₂×D₇	1	2	yes	—
121	#431	—	2	1	no	—
122	#431	—	7	2	no	—
123	#431	—	14	1	no	—
124	#431	Dic₇	1	2	yes	—
125	#423	—	2	1	no	—
126	#423	—	14	1	no	—
127	C₂×C₄⋊C₈	—	14	1	no	—
128	#407	—	7	2	no	—
129	#407	Dic₇	1	2	yes	—
130	C₂×C₂²⋊C₈	—	7	2	no	—
131	C₄×M₁₆	—	2	1	no	—
132	C₄×M₁₆	—	14	1	no	—
133	C₂×#97	—	14	1	no	—
134	C₂³×C₇	—	2	2	no	—
135	C₂³×C₇	—	4	1	no	—
136	C₂³×C₇	—	4	1	no	—
137	C₂³×C₇	#95	1	1	yes	characteristic
138	C₂³×C₇	#95	1	1	yes	characteristic
139	C₂³×C₇	C₂×C₄²	1	1	yes	characteristic
140	C₂×C₄×C₇	—	2	2	no	—
141	C₂×C₄×C₇	—	2	2	no	—
142	C₂×C₄×C₇	—	2	2	no	—
143	C₂×C₄×C₇	—	2	2	no	—
144	C₂×C₄×C₇	—	2	2	no	—
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	—	4	1	no	—
149	C₂×C₄×C₇	—	4	1	no	—
150	C₂×C₄×C₇	—	4	1	no	—
151	C₂×C₄×C₇	—	4	1	no	—
152	C₂×C₄×C₇	—	4	1	no	—
153	C₂×C₄×C₇	—	4	1	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	#95	1	1	yes	characteristic
159	C₂×C₄×C₇	#95	1	1	yes	characteristic
160	C₂×C₄×C₇	#95	1	1	yes	characteristic
161	C₂×C₄×C₇	#95	1	1	yes	characteristic
162	C₂×C₄×C₇	#95	1	1	yes	characteristic
163	C₂×C₄×C₇	#95	1	1	yes	characteristic
164	C₂×C₄×C₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
165	C₂×C₄×C₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
166	C₂×C₄×C₇	C₂×C₂²⋊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	C₇×C₈	—	4	4	no	—
171	C₇⋊C₈	—	4	4	no	—
172	C₇⋊C₈	—	4	4	no	—
173	C₂³×C₄	C₄×D₇	1	1	yes	characteristic
174	C₂³×C₄	C₄×D₇	1	1	yes	characteristic
175	C₂³×C₄	C₂×Dic₇	1	1	yes	characteristic
176	C₂×M₁₆	—	2	2	no	—
177	C₂×M₁₆	—	14	2	no	—
178	C₂²×C₈	—	14	2	no	—
179	C₂×C₄²	Dic₁₄	1	1	yes	characteristic
180	C₂×C₄²	Dic₁₄	1	1	yes	characteristic
181	C₂×C₄²	C₄×D₇	1	1	yes	characteristic
182	C₂×C₄²	C₄×D₇	1	1	yes	characteristic
183	C₂×C₄²	D₂₈	1	1	yes	characteristic
184	C₂×C₄²	D₂₈	1	1	yes	characteristic
185	C₂×C₄²	C₂×Dic₇	1	1	yes	characteristic
186	C₂×C₄²	C₂²⋊D₇	1	1	yes	characteristic
187	C₂×C₄²	C₂²⋊D₇	1	1	yes	characteristic
188	C₂×C₄²	C₂²⋊D₇	1	1	yes	characteristic
189	C₂×C₄²	C₂²⋊D₇	1	1	yes	characteristic
190	C₂×C₄²	C₂²×D₇	1	1	yes	characteristic
191	C₄⋊C₈	—	2	2	no	—
192	C₄⋊C₈	—	2	2	no	—
193	C₄⋊C₈	—	14	2	no	—
194	C₄⋊C₈	—	14	2	no	—
195	C₄⋊C₈	—	14	2	no	—
196	C₄⋊C₈	—	14	2	no	—
197	C₂²⋊C₈	—	2	2	no	—
198	C₂²⋊C₈	—	7	4	no	—
199	C₂²⋊C₈	—	7	4	no	—
200	C₂²⋊C₈	—	14	2	no	—
201	C₂²⋊C₈	—	14	2	no	—
202	C₂²⋊C₈	C₂×Dic₇	1	4	yes	—
203	#97	—	2	2	no	—
204	#97	—	14	2	no	—
205	#97	—	14	2	no	—
206	#97	—	14	2	no	—
207	C₄×C₈	—	2	2	no	—
208	C₄×C₈	—	14	2	no	—
209	C₂²×C₇	—	2	1	no	—
210	C₂²×C₇	—	2	1	no	—
211	C₂²×C₇	—	2	1	no	—
212	C₂²×C₇	—	2	1	no	—
213	C₂²×C₇	—	2	1	no	—
214	C₂²×C₇	—	2	1	no	—
215	C₂²×C₇	—	2	1	no	—
216	C₂²×C₇	—	2	1	no	—
217	C₂²×C₇	—	4	4	no	—
218	C₂²×C₇	#344	1	2	yes	—
219	C₂²×C₇	C₂×#95	1	1	yes	characteristic
220	C₄×C₇	—	2	1	no	—
221	C₄×C₇	—	2	1	no	—
222	C₄×C₇	—	2	1	no	—
223	C₄×C₇	—	2	1	no	—
224	C₄×C₇	—	2	1	no	—
225	C₄×C₇	—	2	1	no	—
226	C₄×C₇	—	2	1	no	—
227	C₄×C₇	—	2	1	no	—
228	C₄×C₇	—	2	1	no	—
229	C₄×C₇	—	2	1	no	—
230	C₄×C₇	#344	1	2	yes	—
231	C₄×C₇	#344	1	2	yes	—
232	C₂⁴	C₄×Dic₇	1	1	yes	characteristic
233	C₂²×C₄	—	4	2	no	—
234	C₂²×C₄	—	4	2	no	—
235	C₂²×C₄	—	4	2	no	—
236	C₂²×C₄	C₄×Dic₇	1	1	yes	characteristic
237	C₂²×C₄	C₄×Dic₇	1	1	yes	characteristic
238	C₂²×C₄	C₄×Dic₇	1	1	yes	characteristic
239	C₂²×C₄	Dic₇⋊C₄	1	1	yes	characteristic
240	C₂²×C₄	Dic₇⋊C₄	1	1	yes	characteristic
241	C₂²×C₄	Dic₇⋊C₄	1	1	yes	characteristic
242	C₂²×C₄	Dic₇⋊C₄	1	1	yes	characteristic
243	C₂²×C₄	C₄⋊Dic₇	1	1	yes	characteristic
244	C₂²×C₄	C₄⋊Dic₇	1	1	yes	characteristic
245	C₂²×C₄	(C₂×D₇)⋊C₄	1	1	yes	characteristic
246	C₂²×C₄	(C₂×D₇)⋊C₄	1	1	yes	characteristic
247	C₂²×C₄	(C₂×D₇)⋊C₄	1	1	yes	characteristic
248	C₂²×C₄	(C₂×D₇)⋊C₄	1	1	yes	characteristic
249	C₂²×C₄	C₂²⋊Dic₇	1	1	yes	characteristic
250	C₂²×C₄	C₂²⋊Dic₇	1	1	yes	characteristic
251	C₂²×C₄	C₂×C₄×D₇	1	1	yes	characteristic
252	C₂²×C₄	C₂×C₄×D₇	1	1	yes	characteristic
253	C₂²×C₄	C₂²×Dic₇	1	1	yes	characteristic
254	M₁₆	—	4	4	no	—
255	M₁₆	—	28	4	no	—
256	C₂×C₈	—	2	4	no	—
257	C₂×C₈	—	14	4	no	—
258	C₂×C₈	—	14	4	no	—
259	C₂×C₈	—	28	4	no	—
260	C₄²	Dic₇⋊C₄	1	1	yes	characteristic
261	C₄²	Dic₇⋊C₄	1	1	yes	characteristic
262	C₄²	Dic₇⋊C₄	1	1	yes	characteristic
263	C₄²	Dic₇⋊C₄	1	1	yes	characteristic
264	C₄²	C₄⋊Dic₇	1	1	yes	characteristic
265	C₄²	C₄⋊Dic₇	1	1	yes	characteristic
266	C₄²	(C₂×D₇)⋊C₄	1	1	yes	characteristic
267	C₄²	(C₂×D₇)⋊C₄	1	1	yes	characteristic
268	C₄²	(C₂×D₇)⋊C₄	1	1	yes	characteristic
269	C₄²	(C₂×D₇)⋊C₄	1	1	yes	characteristic
270	C₄²	C₂²⋊Dic₇	1	1	yes	characteristic
271	C₄²	C₂²⋊Dic₇	1	1	yes	characteristic
272	C₄²	C₂×Dic₁₄	1	1	yes	characteristic
273	C₄²	C₂×D₂₈	1	1	yes	characteristic
274	C₄²	C₂×C₂²⋊D₇	1	1	yes	characteristic
275	C₄²	C₂×C₂²⋊D₇	1	1	yes	characteristic
276	C₂×C₇	—	2	2	no	—
277	C₂×C₇	—	4	1	no	—
278	C₂×C₇	—	4	1	no	—
279	C₂×C₇	C₂×#344	1	1	yes	characteristic
280	C₂×C₇	C₂×#344	1	1	yes	characteristic
281	C₂×C₇	C₂×#344	1	1	yes	characteristic
282	C₂³	—	2	2	no	—
283	C₂³	—	4	1	no	—
284	C₂³	—	4	1	no	—
285	C₂³	C₇⋊#95	1	1	yes	characteristic
286	C₂³	C₇⋊#95	1	1	yes	characteristic
287	C₂³	C₂×C₄×Dic₇	1	1	yes	characteristic
288	C₂×C₄	—	2	2	no	—
289	C₂×C₄	—	2	2	no	—
290	C₂×C₄	—	2	2	no	—
291	C₂×C₄	—	2	2	no	—
292	C₂×C₄	—	2	2	no	—
293	C₂×C₄	—	2	2	no	—
294	C₂×C₄	—	2	2	no	—
295	C₂×C₄	—	2	2	no	—
296	C₂×C₄	—	4	1	no	—
297	C₂×C₄	—	4	1	no	—
298	C₂×C₄	—	4	1	no	—
299	C₂×C₄	—	4	1	no	—
300	C₂×C₄	—	4	1	no	—
301	C₂×C₄	—	4	1	no	—
302	C₂×C₄	—	4	1	no	—
303	C₂×C₄	—	4	1	no	—
304	C₂×C₄	—	4	1	no	—
305	C₂×C₄	—	4	1	no	—
306	C₂×C₄	C₇⋊#95	1	1	yes	characteristic
307	C₂×C₄	C₇⋊#95	1	1	yes	characteristic
308	C₂×C₄	C₇⋊#95	1	1	yes	characteristic
309	C₂×C₄	C₇⋊#95	1	1	yes	characteristic
310	C₂×C₄	C₇⋊#95	1	1	yes	characteristic
311	C₂×C₄	C₇⋊#95	1	1	yes	characteristic
312	C₂×C₄	C₂×Dic₇⋊C₄	1	1	yes	characteristic
313	C₂×C₄	C₂×Dic₇⋊C₄	1	1	yes	characteristic
314	C₂×C₄	C₂×C₄⋊Dic₇	1	1	yes	characteristic
315	C₂×C₄	C₂×(C₂×D₇)⋊C₄	1	1	yes	characteristic
316	C₂×C₄	C₂×(C₂×D₇)⋊C₄	1	1	yes	characteristic
317	C₂×C₄	C₂×C₂²⋊Dic₇	1	1	yes	characteristic
318	C₈	—	4	4	no	—
319	C₈	—	28	4	no	—
320	C₈	—	28	4	no	—
321	C₇	#8124	1	1	yes	characteristic, complemented, Sylow
322	C₂²	—	2	1	no	—
323	C₂²	—	2	1	no	—
324	C₂²	—	2	1	no	—
325	C₂²	—	2	1	no	—
326	C₂²	—	2	1	no	—
327	C₂²	—	2	1	no	—
328	C₂²	—	2	1	no	—
329	C₂²	—	2	1	no	—
330	C₂²	—	4	4	no	—
331	C₂²	#89328	1	2	yes	—
332	C₂²	C₂×C₇⋊#95	1	1	yes	characteristic
333	C₄	—	2	1	no	—
334	C₄	—	2	1	no	—
335	C₄	—	2	1	no	—
336	C₄	—	2	1	no	—
337	C₄	—	2	1	no	—
338	C₄	—	2	1	no	—
339	C₄	—	2	1	no	—
340	C₄	—	2	1	no	—
341	C₄	—	2	1	no	—
342	C₄	—	2	1	no	—
343	C₄	#89351	1	2	yes	—
344	C₄	#89354	1	2	yes	—
345	C₂	—	2	2	no	—
346	C₂	—	4	1	no	—
347	C₂	—	4	1	no	—
348	C₂	C₂×#89328	1	1	yes	characteristic
349	C₂	C₂×#89351	1	1	yes	characteristic
350	C₂	C₂×#89354	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₇⋊#95
commutator	C₂³×C₇	C₂×C₄²
Frattini	C₂×C₄²	C₂²×D₇
Fitting	C₇×#2920	C₂
socle	C₂²×C₇	C₂×#95
prime radical	C₂⁴×C₇	C₄²
cosocle	C₂×C₄²×C₇	C₂³
hypercenter	#2920	D₇
hypocenter	C₇	#8124
LCS	G ▷ C₂³×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. C₇⋊#1308	C₂	semidirect
321. C₇	#8124	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
#11739494	2
C₂×#89698	2
C₇⋊#1308	2
#8124	7