Subgroups — #13380450

Order 1792.

604 conjugacy classes of subgroups.

The subgroup lattice has 376 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×C₇×#125	C₂	1	1	yes	maximal, characteristic
2	#11727239	C₂	1	2	yes	maximal
3	#11727238	C₂	1	2	yes	maximal
4	#11727237	C₂	1	2	yes	maximal
5	#11727237	C₂	1	2	yes	maximal
6	#11727236	C₂	1	2	yes	maximal
7	#11727236	C₂	1	2	yes	maximal
8	#11727146	C₂	1	1	yes	maximal, characteristic
9	#11727146	C₂	1	1	yes	maximal, characteristic
10	C₂×C₇×#125	C₄	1	1	yes	characteristic
11	C₇×#389	C₄	1	1	yes	characteristic
12	C₇×#389	C₄	1	1	yes	characteristic
13	C₇×#387	C₄	1	2	yes	—
14	C₇×#387	C₄	1	2	yes	—
15	C₇×#382	C₄	1	1	yes	characteristic
16	C₄×C₇×C₄⋊C₄	C₂²	1	1	yes	characteristic
17	C₄×C₇×C₄⋊C₄	C₂²	1	2	yes	—
18	C₄×C₇×C₄⋊C₄	C₂²	1	1	yes	characteristic
19	C₄×C₇×C₄⋊C₄	C₂²	1	2	yes	—
20	C₄³×C₇	C₂²	1	1	yes	characteristic
21	C₂×C₄⋊(C₇⋊C₈)	C₂²	1	1	yes	characteristic
22	C₂×C₄⋊(C₇⋊C₈)	C₂²	1	1	yes	characteristic
23	C₂×C₄⋊(C₇⋊C₈)	C₂²	1	1	yes	characteristic
24	C₂×C₄⋊(C₇⋊C₈)	C₂²	1	1	yes	characteristic
25	C₂×C₄⋊(C₇⋊C₈)	C₂²	1	2	yes	—
26	C₂×C₄⋊(C₇⋊C₈)	C₂²	1	2	yes	—
27	C₂×#6377	C₂²	1	2	yes	—
28	C₂×C₄×C₇⋊C₈	C₂²	1	2	yes	—
29	#89319	C₂²	1	2	yes	—
30	#89319	C₂²	1	2	yes	—
31	#89319	C₂²	1	2	yes	—
32	#89319	C₂²	1	2	yes	—
33	#89319	C₂²	1	2	yes	—
34	#89319	C₂²	1	2	yes	—
35	#89319	C₂²	1	2	yes	—
36	#89319	C₂²	1	2	yes	—
37	#23371	—	7	1	no	maximal, Sylow
38	C₇×#125	—	2	4	no	—
39	C₂×C₇×C₄⋊C₄	C₂×C₄	1	2	yes	—
40	C₂×C₇×C₄⋊C₄	C₂×C₄	1	2	yes	—
41	C₂×C₇×C₄⋊C₄	C₂×C₄	1	2	yes	—
42	C₂×C₇×C₄⋊C₄	C₂×C₄	1	2	yes	—
43	C₂×C₇×C₄⋊C₄	C₂×C₄	1	1	yes	characteristic
44	C₂×C₇×C₄⋊C₄	C₂×C₄	1	1	yes	characteristic
45	C₂×C₇×C₄⋊C₄	C₂×C₄	1	1	yes	characteristic
46	C₂×C₇×C₄⋊C₄	C₂×C₄	1	1	yes	characteristic
47	C₂×C₄²×C₇	C₂×C₄	1	1	yes	characteristic
48	C₂×C₄²×C₇	C₂×C₄	1	1	yes	characteristic
49	C₂×C₄²×C₇	C₂×C₄	1	1	yes	characteristic
50	C₂×C₄²×C₇	C₂×C₄	1	1	yes	characteristic
51	C₂×C₄²×C₇	C₂³	1	2	yes	—
52	C₂×C₄²×C₇	C₂³	1	2	yes	—
53	C₂×C₄²×C₇	C₂³	1	1	yes	characteristic
54	C₂×C₄²×C₇	C₂³	1	1	yes	characteristic
55	C₂×C₄²×C₇	C₂³	1	1	yes	characteristic
56	C₂²×C₇⋊C₈	C₂³	1	2	yes	—
57	C₂²×C₇⋊C₈	C₂³	1	2	yes	—
58	C₂²×C₇⋊C₈	C₂³	1	2	yes	—
59	C₂²×C₇⋊C₈	C₂³	1	2	yes	—
60	C₇×#95	C₂×C₄	1	2	yes	—
61	C₇×#95	C₂×C₄	1	2	yes	—
62	C₇×#95	C₂×C₄	1	2	yes	—
63	C₇×#95	C₂×C₄	1	2	yes	—
64	C₇×#95	C₂×C₄	1	2	yes	—
65	C₇×#95	C₂×C₄	1	2	yes	—
66	C₄⋊(C₇⋊C₈)	—	4	1	no	—
67	C₄⋊(C₇⋊C₈)	—	4	2	no	—
68	C₄⋊(C₇⋊C₈)	—	4	1	no	—
69	C₄⋊(C₇⋊C₈)	—	4	2	no	—
70	C₄⋊(C₇⋊C₈)	—	4	1	no	—
71	C₄⋊(C₇⋊C₈)	—	4	1	no	—
72	#6377	—	4	2	no	—
73	C₄×C₇⋊C₈	—	4	2	no	—
74	C₄×#125	D₇	1	1	yes	characteristic
75	#1940	—	7	2	no	—
76	#1939	—	7	2	no	—
77	#1917	—	7	2	no	—
78	#1917	—	7	2	no	—
79	#1916	—	7	2	no	—
80	#1916	—	7	2	no	—
81	#1831	—	7	1	no	—
82	#1831	—	7	1	no	—
83	C₂²×C₄×C₇	C₂²×C₄	1	1	yes	characteristic
84	C₂²×C₄×C₇	C₂²×C₄	1	2	yes	—
85	C₂²×C₄×C₇	C₂²×C₄	1	2	yes	—
86	C₂²×C₄×C₇	C₂²×C₄	1	1	yes	characteristic
87	C₂²×C₄×C₇	C₂²×C₄	1	1	yes	characteristic
88	C₂²×C₄×C₇	C₂²×C₄	1	2	yes	—
89	C₂²×C₄×C₇	C₂²×C₄	1	2	yes	—
90	C₂²×C₄×C₇	C₂²×C₄	1	1	yes	characteristic
91	C₂²×C₄×C₇	C₂²×C₄	1	1	yes	characteristic
92	C₂²×C₄×C₇	C₂²×C₄	1	1	yes	characteristic
93	C₂²×C₄×C₇	C₂⁴	1	1	yes	characteristic
94	C₇×C₄⋊C₄	—	2	4	no	—
95	C₇×C₄⋊C₄	—	2	4	no	—
96	C₇×C₄⋊C₄	—	2	2	no	—
97	C₇×C₄⋊C₄	—	2	4	no	—
98	C₇×C₄⋊C₄	—	2	4	no	—
99	C₇×C₄⋊C₄	—	4	1	no	—
100	C₇×C₄⋊C₄	—	4	1	no	—
101	C₇×C₄⋊C₄	M₁₆	1	4	yes	—
102	C₄²×C₇	—	2	2	no	—
103	C₄²×C₇	—	2	2	no	—
104	C₄²×C₇	—	2	2	no	—
105	C₄²×C₇	—	2	2	no	—
106	C₄²×C₇	—	2	2	no	—
107	C₄²×C₇	—	2	1	no	—
108	C₄²×C₇	—	2	1	no	—
109	C₄²×C₇	—	2	1	no	—
110	C₄²×C₇	—	2	1	no	—
111	C₄²×C₇	—	2	2	no	—
112	C₄²×C₇	—	2	1	no	—
113	C₄²×C₇	—	2	1	no	—
114	C₄²×C₇	—	2	2	no	—
115	C₄²×C₇	—	4	1	no	—
116	C₂×C₇⋊C₈	—	2	2	no	—
117	C₂×C₇⋊C₈	—	2	2	no	—
118	C₂×C₇⋊C₈	—	2	2	no	—
119	C₂×C₇⋊C₈	—	2	2	no	—
120	C₂×C₇⋊C₈	—	2	2	no	—
121	C₂×C₇⋊C₈	—	2	2	no	—
122	C₂×C₇⋊C₈	—	2	2	no	—
123	C₂×C₇⋊C₈	—	2	2	no	—
124	C₂×C₇⋊C₈	—	2	2	no	—
125	C₂×C₇⋊C₈	—	2	2	no	—
126	C₂×C₇⋊C₈	—	2	2	no	—
127	C₂×C₇⋊C₈	—	2	2	no	—
128	C₂×#125	Dic₇	1	1	yes	characteristic
129	C₂×C₄⋊C₈	—	7	1	no	—
130	C₂×C₄⋊C₈	—	7	1	no	—
131	C₂×C₄⋊C₈	—	7	1	no	—
132	C₂×C₄⋊C₈	—	7	1	no	—
133	C₂×C₄⋊C₈	—	7	2	no	—
134	C₂×C₄⋊C₈	—	7	2	no	—
135	C₂×#97	—	7	2	no	—
136	C₂×C₄×C₈	—	7	2	no	—
137	#389	Dic₇	1	1	yes	characteristic
138	#389	Dic₇	1	1	yes	characteristic
139	#387	Dic₇	1	2	yes	—
140	#387	Dic₇	1	2	yes	—
141	#382	Dic₇	1	1	yes	characteristic
142	C₄×C₄⋊C₄	C₂×D₇	1	1	yes	characteristic
143	C₄×C₄⋊C₄	C₂×D₇	1	2	yes	—
144	C₄×C₄⋊C₄	C₂×D₇	1	1	yes	characteristic
145	C₄×C₄⋊C₄	C₂×D₇	1	2	yes	—
146	C₄³	C₂×D₇	1	1	yes	characteristic
147	#336	—	7	2	no	—
148	#336	—	7	2	no	—
149	#336	—	7	2	no	—
150	#336	—	7	2	no	—
151	#336	—	7	2	no	—
152	#336	—	7	2	no	—
153	#336	—	7	2	no	—
154	#336	—	7	2	no	—
155	C₂³×C₇	C₂³×C₄	1	1	yes	characteristic
156	C₂×C₄×C₇	—	2	1	no	—
157	C₂×C₄×C₇	—	2	2	no	—
158	C₂×C₄×C₇	—	2	1	no	—
159	C₂×C₄×C₇	—	2	1	no	—
160	C₂×C₄×C₇	—	2	2	no	—
161	C₂×C₄×C₇	—	2	1	no	—
162	C₂×C₄×C₇	—	2	1	no	—
163	C₂×C₄×C₇	—	2	1	no	—
164	C₂×C₄×C₇	—	2	1	no	—
165	C₂×C₄×C₇	—	2	1	no	—
166	C₂×C₄×C₇	—	2	2	no	—
167	C₂×C₄×C₇	—	2	1	no	—
168	C₂×C₄×C₇	—	2	2	no	—
169	C₂×C₄×C₇	—	2	2	no	—
170	C₂×C₄×C₇	—	2	1	no	—
171	C₂×C₄×C₇	—	2	1	no	—
172	C₂×C₄×C₇	—	2	2	no	—
173	C₂×C₄×C₇	—	2	1	no	—
174	C₂×C₄×C₇	—	2	1	no	—
175	C₂×C₄×C₇	—	2	1	no	—
176	C₂×C₄×C₇	—	2	1	no	—
177	C₂×C₄×C₇	—	2	2	no	—
178	C₂×C₄×C₇	—	2	2	no	—
179	C₂×C₄×C₇	—	2	2	no	—
180	C₂×C₄×C₇	—	2	2	no	—
181	C₂×C₄×C₇	—	2	2	no	—
182	C₂×C₄×C₇	C₂×M₁₆	1	2	yes	—
183	C₂×C₄×C₇	C₂×M₁₆	1	4	yes	—
184	C₂×C₄×C₇	#131	1	2	yes	—
185	C₂×C₄×C₇	#131	1	2	yes	—
186	C₂×C₄×C₇	H(5,𝔽₂)	1	1	yes	characteristic
187	C₂×C₄×C₇	H(5,𝔽₂)	1	1	yes	characteristic
188	C₂×C₄×C₇	2₋¹⁺⁴	1	1	yes	characteristic
189	C₂×C₄×C₇	2₋¹⁺⁴	1	1	yes	characteristic
190	C₂×C₄×C₇	2₋¹⁺⁴	1	1	yes	characteristic
191	C₂×C₄×C₇	2₋¹⁺⁴	1	1	yes	characteristic
192	C₇⋊C₈	—	4	2	no	—
193	C₇⋊C₈	—	4	2	no	—
194	C₇⋊C₈	—	4	2	no	—
195	C₇⋊C₈	—	4	2	no	—
196	C₂²×C₈	—	7	2	no	—
197	C₂²×C₈	—	7	2	no	—
198	C₂²×C₈	—	7	2	no	—
199	C₂²×C₈	—	7	2	no	—
200	#125	—	2	4	no	—
201	C₂×C₄⋊C₄	C₂×Dic₇	1	2	yes	—
202	C₂×C₄⋊C₄	C₂×Dic₇	1	2	yes	—
203	C₂×C₄⋊C₄	C₂×Dic₇	1	1	yes	characteristic
204	C₂×C₄⋊C₄	C₂×Dic₇	1	2	yes	—
205	C₂×C₄⋊C₄	C₂×Dic₇	1	2	yes	—
206	C₂×C₄⋊C₄	C₂×Dic₇	1	1	yes	characteristic
207	C₂×C₄⋊C₄	C₂×Dic₇	1	1	yes	characteristic
208	C₂×C₄⋊C₄	C₂×Dic₇	1	1	yes	characteristic
209	C₂×C₄²	C₂×Dic₇	1	1	yes	characteristic
210	C₂×C₄²	C₂×Dic₇	1	1	yes	characteristic
211	C₂×C₄²	C₂×Dic₇	1	1	yes	characteristic
212	C₂×C₄²	C₂×Dic₇	1	1	yes	characteristic
213	C₂×C₄²	C₂²×D₇	1	1	yes	characteristic
214	C₂×C₄²	C₂²×D₇	1	1	yes	characteristic
215	C₂×C₄²	C₂²×D₇	1	1	yes	characteristic
216	C₂×C₄²	C₂²×D₇	1	2	yes	—
217	C₂×C₄²	C₂²×D₇	1	2	yes	—
218	C₄⋊C₈	—	28	2	no	—
219	C₄⋊C₈	—	28	2	no	—
220	C₄⋊C₈	—	28	1	no	—
221	C₄⋊C₈	—	28	1	no	—
222	C₄⋊C₈	—	28	1	no	—
223	C₄⋊C₈	—	28	1	no	—
224	#97	—	28	2	no	—
225	C₄×C₈	—	28	2	no	—
226	#95	C₂×Dic₇	1	2	yes	—
227	#95	C₂×Dic₇	1	2	yes	—
228	#95	C₂×Dic₇	1	2	yes	—
229	#95	C₂×Dic₇	1	2	yes	—
230	#95	C₂×Dic₇	1	2	yes	—
231	#95	C₂×Dic₇	1	2	yes	—
232	C₂²×C₇	#519	1	1	yes	characteristic
233	C₂²×C₇	#520	1	1	yes	characteristic
234	C₂²×C₇	#520	1	1	yes	characteristic
235	C₂²×C₇	C₂²×M₁₆	1	1	yes	characteristic
236	C₂²×C₇	C₂×#131	1	1	yes	characteristic
237	C₂²×C₇	C₂×#131	1	1	yes	characteristic
238	C₂²×C₇	#568	1	1	yes	characteristic
239	C₄×C₇	—	2	2	no	—
240	C₄×C₇	—	2	4	no	—
241	C₄×C₇	—	2	2	no	—
242	C₄×C₇	—	2	2	no	—
243	C₄×C₇	—	4	1	no	—
244	C₄×C₇	—	4	1	no	—
245	C₄×C₇	—	4	1	no	—
246	C₄×C₇	—	4	2	no	—
247	C₄×C₇	—	4	2	no	—
248	C₄×C₇	—	4	2	no	—
249	C₄×C₇	#562	1	1	yes	characteristic
250	C₄×C₇	#563	1	1	yes	characteristic
251	C₄×C₇	#563	1	1	yes	characteristic
252	C₄×C₇	#564	1	1	yes	characteristic
253	C₂²×C₄	C₂²×Dic₇	1	1	yes	characteristic
254	C₂²×C₄	C₂²×Dic₇	1	2	yes	—
255	C₂²×C₄	C₂²×Dic₇	1	2	yes	—
256	C₂²×C₄	C₂²×Dic₇	1	1	yes	characteristic
257	C₂²×C₄	C₂²×Dic₇	1	1	yes	characteristic
258	C₂²×C₄	C₂²×Dic₇	1	2	yes	—
259	C₂²×C₄	C₂²×Dic₇	1	2	yes	—
260	C₂²×C₄	C₂²×Dic₇	1	1	yes	characteristic
261	C₂²×C₄	C₂²×Dic₇	1	1	yes	characteristic
262	C₂²×C₄	C₂²×Dic₇	1	1	yes	characteristic
263	C₂²×C₄	C₂³×D₇	1	1	yes	characteristic
264	C₂×C₈	—	14	2	no	—
265	C₂×C₈	—	14	2	no	—
266	C₂×C₈	—	14	2	no	—
267	C₂×C₈	—	14	2	no	—
268	C₂×C₈	—	14	2	no	—
269	C₂×C₈	—	14	2	no	—
270	C₂×C₈	—	14	2	no	—
271	C₂×C₈	—	14	2	no	—
272	C₂×C₈	—	14	2	no	—
273	C₂×C₈	—	14	2	no	—
274	C₂×C₈	—	14	2	no	—
275	C₂×C₈	—	14	2	no	—
276	C₄⋊C₄	—	2	4	no	—
277	C₄⋊C₄	—	2	4	no	—
278	C₄⋊C₄	—	2	2	no	—
279	C₄⋊C₄	—	2	4	no	—
280	C₄⋊C₄	—	2	4	no	—
281	C₄⋊C₄	—	4	1	no	—
282	C₄⋊C₄	—	4	1	no	—
283	C₄⋊C₄	#1136	1	4	yes	—
284	C₄²	—	2	2	no	—
285	C₄²	—	2	2	no	—
286	C₄²	—	2	2	no	—
287	C₄²	—	2	1	no	—
288	C₄²	—	2	1	no	—
289	C₄²	—	2	2	no	—
290	C₄²	—	2	2	no	—
291	C₄²	—	2	1	no	—
292	C₄²	—	2	1	no	—
293	C₄²	—	2	1	no	—
294	C₄²	—	2	1	no	—
295	C₄²	—	2	2	no	—
296	C₄²	—	2	2	no	—
297	C₄²	—	4	1	no	—
298	C₂×C₇	#2382	1	1	yes	characteristic
299	C₂×C₇	#2983	1	1	yes	characteristic
300	C₂×C₇	#2986	1	1	yes	characteristic
301	C₂×C₇	#2990	1	1	yes	characteristic
302	C₂×C₇	#2991	1	1	yes	characteristic
303	C₂×C₇	#2992	1	1	yes	characteristic
304	C₂×C₇	#2992	1	1	yes	characteristic
305	C₂³	C₂³×Dic₇	1	1	yes	characteristic
306	C₂×C₄	—	2	1	no	—
307	C₂×C₄	—	2	2	no	—
308	C₂×C₄	—	2	1	no	—
309	C₂×C₄	—	2	1	no	—
310	C₂×C₄	—	2	1	no	—
311	C₂×C₄	—	2	2	no	—
312	C₂×C₄	—	2	1	no	—
313	C₂×C₄	—	2	1	no	—
314	C₂×C₄	—	2	1	no	—
315	C₂×C₄	—	2	1	no	—
316	C₂×C₄	—	2	2	no	—
317	C₂×C₄	—	2	1	no	—
318	C₂×C₄	—	2	1	no	—
319	C₂×C₄	—	2	1	no	—
320	C₂×C₄	—	2	1	no	—
321	C₂×C₄	—	2	1	no	—
322	C₂×C₄	—	2	2	no	—
323	C₂×C₄	—	2	2	no	—
324	C₂×C₄	—	2	2	no	—
325	C₂×C₄	—	2	2	no	—
326	C₂×C₄	—	2	2	no	—
327	C₂×C₄	—	2	1	no	—
328	C₂×C₄	—	2	2	no	—
329	C₂×C₄	—	2	1	no	—
330	C₂×C₄	—	2	2	no	—
331	C₂×C₄	—	2	2	no	—
332	C₂×C₄	C₂×#1136	1	2	yes	—
333	C₂×C₄	C₂×#1136	1	4	yes	—
334	C₂×C₄	#6511	1	2	yes	—
335	C₂×C₄	#6511	1	2	yes	—
336	C₂×C₄	#6551	1	1	yes	characteristic
337	C₂×C₄	#6551	1	1	yes	characteristic
338	C₂×C₄	#6553	1	1	yes	characteristic
339	C₂×C₄	#6553	1	1	yes	characteristic
340	C₂×C₄	#6554	1	1	yes	characteristic
341	C₂×C₄	#6554	1	1	yes	characteristic
342	C₈	—	28	2	no	—
343	C₈	—	28	2	no	—
344	C₈	—	28	2	no	—
345	C₈	—	28	2	no	—
346	C₇	#23371	1	1	yes	characteristic, complemented, Sylow
347	C₂²	C₂²×#1136	1	1	yes	characteristic
348	C₂²	#90503	1	1	yes	characteristic
349	C₂²	C₂×#6511	1	1	yes	characteristic
350	C₂²	C₂×#6511	1	1	yes	characteristic
351	C₂²	#90510	1	1	yes	characteristic
352	C₂²	#90518	1	1	yes	characteristic
353	C₂²	#90518	1	1	yes	characteristic
354	C₄	—	2	4	no	—
355	C₄	—	2	2	no	—
356	C₄	—	2	2	no	—
357	C₄	—	2	2	no	—
358	C₄	—	4	1	no	—
359	C₄	—	4	1	no	—
360	C₄	—	4	2	no	—
361	C₄	—	4	1	no	—
362	C₄	—	4	2	no	—
363	C₄	—	4	2	no	—
364	C₄	C₇⋊#564	1	1	yes	characteristic
365	C₄	#90390	1	1	yes	characteristic
366	C₄	#90390	1	1	yes	characteristic
367	C₄	#90391	1	1	yes	characteristic
368	C₂	#11739881	1	1	yes	characteristic
369	C₂	#11739967	1	1	yes	characteristic
370	C₂	#11739968	1	1	yes	characteristic
371	C₂	#11740472	1	1	yes	characteristic
372	C₂	#11740575	1	1	yes	characteristic
373	C₂	#11740575	1	1	yes	characteristic
374	C₂	#11740777	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂³×D₇
commutator, socle, prime radical	C₂³×C₇	C₂³×C₄
Fitting	C₄×C₇×#125	C₂
cosocle	C₂²×C₄×C₇	C₂⁴
hypercenter	C₄×#125	D₇
hypocenter	C₇	#23371
LCS	G ▷ C₂³×C₇ ▷ C₇
UCS	C₁ ◁ C₂²×C₄ ◁ C₄×#125

Complements

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

N	H ≅ G/N	type
346. C₇	#23371	semidirect

Maximal subgroups

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

subgroup	index
C₄×C₇×#125	2
#11727239	2
#11727238	2
#11727237	2
#11727236	2
#11727146	2
#23371	7