Subgroups — C₇×#36639 #13213855

Order 1792.

2782 conjugacy classes of subgroups.

The subgroup lattice has 354 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₇×#533	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₇×#532	C₂	1	1	yes	maximal, characteristic
3	C₇×#2862	C₂	1	2	yes	maximal, complemented
4	C₇×#2861	C₂	1	4	yes	maximal, complemented
5	C₇×#2858	C₂	1	2	yes	maximal, complemented
6	C₇×#2850	C₂	1	8	yes	maximal, complemented
7	C₄×C₇×#124	C₂	1	8	yes	maximal, complemented
8	C₇×#2297	C₂	1	4	yes	maximal, complemented
9	C₂×C₄³×C₇	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂²×C₇×#41	C₂²	1	2	yes	—
11	C₇×#533	—	2	8	no	—
12	C₇×#532	—	2	8	no	—
13	C₂×C₇×C₄⋊Q₈	C₂²	1	1	yes	characteristic
14	C₂×C₇×C₄⋊Q₈	C₂²	1	2	yes	—
15	C₂×C₇×#127	C₂²	1	1	yes	characteristic
16	C₂×C₇×#125	C₂²	1	2	yes	—
17	C₂×C₇×#124	C₂²	1	2	yes	—
18	C₂×C₇×#124	C₂²	1	8	yes	—
19	C₂×C₇×C₂²⋊Q₈	C₂²	1	4	yes	—
20	C₂×C₇×#121	C₂²	1	4	yes	—
21	C₂×C₄×C₇×Q₈	C₂²	1	4	yes	—
22	C₂×C₄×C₇×D₄	C₂²	1	4	yes	—
23	C₂×C₇×#117	C₂²	1	2	yes	—
24	C₂²×C₄²×C₇	C₂²	1	2	yes	—
25	C₂²×C₄²×C₇	C₂²	1	4	yes	—
26	C₂²×C₄²×C₇	C₂²	1	1	yes	characteristic
27	C₇×#391	C₂²	1	8	yes	—
28	C₇×#391	C₂²	1	8	yes	—
29	C₇×#390	C₂²	1	8	yes	—
30	C₇×#390	C₂²	1	8	yes	—
31	C₇×#388	C₂²	1	32	yes	—
32	C₇×#386	C₂²	1	16	yes	—
33	C₇×#382	C₂²	1	8	yes	—
34	C₄×C₇×C₂²⋊C₄	C₂²	1	16	yes	—
35	C₄³×C₇	C₂²	1	4	yes	—
36	C₄³×C₇	C₂²	1	2	yes	—
37	C₄³×C₇	C₂²	1	2	yes	—
38	#36639	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
39	C₂×C₇×#41	—	2	8	no	—
40	C₂×C₇×#41	—	2	16	no	—
41	C₂²×C₇×Q₈	C₂³	1	2	yes	—
42	C₂²×C₇×Q₈	C₂³	1	4	yes	—
43	C₂²×C₇×D₄	C₂³	1	4	yes	—
44	C₂²×C₇×D₄	C₂³	1	2	yes	—
45	C₂²×C₇×D₄	C₂³	1	4	yes	—
46	C₂³×C₄×C₇	C₂³	1	4	yes	—
47	C₂³×C₄×C₇	C₂³	1	4	yes	—
48	C₂³×C₄×C₇	C₂³	1	2	yes	—
49	C₂³×C₄×C₇	C₂³	1	1	yes	characteristic
50	C₇×C₄⋊Q₈	—	2	4	no	—
51	C₇×C₄⋊Q₈	—	2	8	no	—
52	C₇×#127	—	2	4	no	—
53	C₇×#125	—	2	8	no	—
54	C₇×#124	—	2	8	no	—
55	C₇×#124	—	2	32	no	—
56	C₇×C₂²⋊Q₈	—	2	16	no	—
57	C₇×#121	—	2	16	no	—
58	C₄×C₇×Q₈	—	2	16	no	—
59	C₄×C₇×D₄	—	2	16	no	—
60	C₇×#117	—	2	8	no	—
61	C₂×C₇×C₄⋊C₄	C₂³	1	4	yes	—
62	C₂×C₇×C₄⋊C₄	C₂³	1	4	yes	—
63	C₂×C₇×C₄⋊C₄	C₂³	1	4	yes	—
64	C₂×C₇×C₄⋊C₄	C₂³	1	8	yes	—
65	C₂×C₇×C₂²⋊C₄	C₂³	1	32	yes	—
66	C₂×C₇×C₂²⋊C₄	C₂³	1	4	yes	—
67	C₂×C₇×C₂²⋊C₄	C₂³	1	8	yes	—
68	C₂×C₄²×C₇	—	2	8	no	—
69	C₂×C₄²×C₇	—	2	16	no	—
70	C₂×C₄²×C₇	D₄	1	8	yes	—
71	C₂×C₄²×C₇	C₂³	1	8	yes	—
72	C₂×C₄²×C₇	C₂³	1	4	yes	—
73	C₂×C₄²×C₇	C₂³	1	4	yes	—
74	C₂×C₄²×C₇	C₂³	1	4	yes	—
75	C₂×C₄²×C₇	C₂³	1	4	yes	—
76	C₂×C₄²×C₇	C₂³	1	1	yes	characteristic
77	C₂×C₄²×C₇	C₂³	1	1	yes	characteristic
78	C₂×C₄²×C₇	C₂³	1	2	yes	—
79	C₂×C₄²×C₇	C₂³	1	4	yes	—
80	C₇×#95	C₂³	1	32	yes	—
81	C₂×#533	C₂×C₇	1	1	yes	characteristic, complemented
82	C₂×#532	C₂×C₇	1	1	yes	characteristic
83	#2862	C₂×C₇	1	2	yes	complemented
84	#2861	C₂×C₇	1	4	yes	complemented
85	#2858	C₂×C₇	1	2	yes	complemented
86	#2850	C₂×C₇	1	8	yes	complemented
87	C₄×#124	C₂×C₇	1	8	yes	complemented
88	#2297	C₂×C₇	1	4	yes	complemented
89	C₂×C₄³	C₂×C₇	1	1	yes	characteristic, complemented
90	C₂⁴×C₇	C₂⁴	1	4	yes	—
91	C₂⁴×C₇	C₂⁴	1	1	yes	characteristic
92	C₇×#41	—	4	32	no	—
93	C₂×C₇×Q₈	—	2	8	no	—
94	C₂×C₇×Q₈	—	2	4	no	—
95	C₂×C₇×Q₈	—	2	16	no	—
96	C₂×C₇×Q₈	—	2	8	no	—
97	C₂×C₇×D₄	—	2	8	no	—
98	C₂×C₇×D₄	—	2	16	no	—
99	C₂×C₇×D₄	—	2	8	no	—
100	C₂×C₇×D₄	—	2	4	no	—
101	C₂×C₇×D₄	—	2	16	no	—
102	C₂×C₇×D₄	—	2	8	no	—
103	C₂²×C₄×C₇	—	2	8	no	—
104	C₂²×C₄×C₇	—	2	16	no	—
105	C₂²×C₄×C₇	—	2	8	no	—
106	C₂²×C₄×C₇	—	2	8	no	—
107	C₂²×C₄×C₇	—	2	16	no	—
108	C₂²×C₄×C₇	C₂×D₄	1	8	yes	—
109	C₂²×C₄×C₇	C₂×D₄	1	4	yes	—
110	C₂²×C₄×C₇	#41	1	8	yes	—
111	C₂²×C₄×C₇	C₂⁴	1	4	yes	—
112	C₂²×C₄×C₇	C₂⁴	1	8	yes	—
113	C₂²×C₄×C₇	C₂⁴	1	2	yes	—
114	C₂²×C₄×C₇	C₂⁴	1	8	yes	—
115	C₂²×C₄×C₇	C₂⁴	1	1	yes	characteristic
116	C₂²×C₄×C₇	C₂⁴	1	2	yes	—
117	C₂²×C₄×C₇	C₂⁴	1	1	yes	characteristic
118	C₇×C₄⋊C₄	—	2	8	no	—
119	C₇×C₄⋊C₄	—	2	8	no	—
120	C₇×C₄⋊C₄	—	2	16	no	—
121	C₇×C₄⋊C₄	—	2	8	no	—
122	C₇×C₂²⋊C₄	—	2	8	no	—
123	C₇×C₂²⋊C₄	—	2	16	no	—
124	C₇×C₂²⋊C₄	—	2	64	no	—
125	C₄²×C₇	—	2	8	no	—
126	C₄²×C₇	—	2	16	no	—
127	C₄²×C₇	—	2	8	no	—
128	C₄²×C₇	—	2	8	no	—
129	C₄²×C₇	—	2	8	no	—
130	C₄²×C₇	C₂×D₄	1	8	yes	—
131	C₄²×C₇	C₂×D₄	1	4	yes	—
132	C₄²×C₇	C₂×D₄	1	4	yes	—
133	C₄²×C₇	#41	1	16	yes	complemented
134	C₂²×#41	C₂²×C₇	1	2	yes	—
135	#533	—	2	8	no	—
136	#532	—	2	8	no	—
137	C₂×C₄⋊Q₈	C₂²×C₇	1	2	yes	—
138	C₂×C₄⋊Q₈	C₂²×C₇	1	1	yes	characteristic
139	C₂×#127	C₂²×C₇	1	1	yes	characteristic
140	C₂×#125	C₂²×C₇	1	2	yes	—
141	C₂×#124	C₂²×C₇	1	8	yes	—
142	C₂×#124	C₂²×C₇	1	2	yes	—
143	C₂×C₂²⋊Q₈	C₂²×C₇	1	4	yes	—
144	C₂×#121	C₂²×C₇	1	4	yes	—
145	C₂×C₄×Q₈	C₂²×C₇	1	4	yes	—
146	C₂×C₄×D₄	C₂²×C₇	1	4	yes	—
147	C₂×#117	C₂²×C₇	1	2	yes	—
148	C₂²×C₄²	C₂²×C₇	1	4	yes	—
149	C₂²×C₄²	C₂²×C₇	1	2	yes	—
150	C₂²×C₄²	C₂²×C₇	1	1	yes	characteristic
151	#391	C₂²×C₇	1	8	yes	—
152	#391	C₂²×C₇	1	8	yes	—
153	#390	C₂²×C₇	1	8	yes	—
154	#390	C₂²×C₇	1	8	yes	—
155	#388	C₂²×C₇	1	32	yes	—
156	#386	C₂²×C₇	1	16	yes	—
157	#382	C₂²×C₇	1	8	yes	—
158	C₄×C₂²⋊C₄	C₂²×C₇	1	16	yes	—
159	C₄³	C₂²×C₇	1	2	yes	—
160	C₄³	C₂²×C₇	1	2	yes	—
161	C₄³	C₂²×C₇	1	4	yes	—
162	C₂³×C₇	—	2	4	no	—
163	C₂³×C₇	—	2	4	no	—
164	C₂³×C₇	—	2	16	no	—
165	C₂³×C₇	—	2	8	no	—
166	C₂³×C₇	C₂²×D₄	1	2	yes	—
167	C₂³×C₇	C₂×#41	1	4	yes	—
168	C₂³×C₇	C₂⁵	1	1	yes	characteristic
169	C₇×Q₈	—	4	8	no	—
170	C₇×Q₈	—	4	16	no	—
171	C₇×D₄	—	4	16	no	—
172	C₇×D₄	—	4	8	no	—
173	C₇×D₄	—	4	16	no	—
174	C₂×C₄×C₇	—	2	32	no	—
175	C₂×C₄×C₇	—	2	4	no	—
176	C₂×C₄×C₇	—	2	8	no	—
177	C₂×C₄×C₇	—	2	8	no	—
178	C₂×C₄×C₇	—	2	16	no	—
179	C₂×C₄×C₇	—	2	8	no	—
180	C₂×C₄×C₇	—	2	4	no	—
181	C₂×C₄×C₇	—	2	16	no	—
182	C₂×C₄×C₇	—	2	8	no	—
183	C₂×C₄×C₇	—	4	16	no	—
184	C₂×C₄×C₇	#124	1	16	yes	complemented
185	C₂×C₄×C₇	C₂²×D₄	1	4	yes	—
186	C₂×C₄×C₇	C₂²×D₄	1	2	yes	—
187	C₂×C₄×C₇	C₂²×D₄	1	2	yes	—
188	C₂×C₄×C₇	C₂²×D₄	1	4	yes	—
189	C₂×C₄×C₇	C₂×#41	1	16	yes	—
190	C₂×C₄×C₇	C₂×#41	1	4	yes	—
191	C₂×C₄×C₇	C₂×#41	1	4	yes	—
192	C₂×C₄×C₇	C₂×#41	1	8	yes	—
193	C₂×#41	—	2	8	no	—
194	C₂×#41	—	2	16	no	—
195	C₂²×Q₈	C₂³×C₇	1	4	yes	—
196	C₂²×Q₈	C₂³×C₇	1	2	yes	—
197	C₂²×D₄	C₂³×C₇	1	4	yes	—
198	C₂²×D₄	C₂³×C₇	1	4	yes	—
199	C₂²×D₄	C₂³×C₇	1	2	yes	—
200	C₂³×C₄	C₂³×C₇	1	4	yes	—
201	C₂³×C₄	C₂³×C₇	1	4	yes	—
202	C₂³×C₄	C₂³×C₇	1	2	yes	—
203	C₂³×C₄	C₂³×C₇	1	1	yes	characteristic
204	C₄⋊Q₈	—	2	4	no	—
205	C₄⋊Q₈	—	2	8	no	—
206	#127	—	2	4	no	—
207	#125	—	2	8	no	—
208	#124	—	2	8	no	—
209	#124	—	2	32	no	—
210	C₂²⋊Q₈	—	2	16	no	—
211	#121	—	2	16	no	—
212	C₄×Q₈	—	2	16	no	—
213	C₄×D₄	—	2	16	no	—
214	#117	—	2	8	no	—
215	C₂×C₄⋊C₄	C₂³×C₇	1	4	yes	—
216	C₂×C₄⋊C₄	C₂³×C₇	1	4	yes	—
217	C₂×C₄⋊C₄	C₂³×C₇	1	8	yes	—
218	C₂×C₄⋊C₄	C₂³×C₇	1	4	yes	—
219	C₂×C₂²⋊C₄	C₂³×C₇	1	4	yes	—
220	C₂×C₂²⋊C₄	C₂³×C₇	1	8	yes	—
221	C₂×C₂²⋊C₄	C₂³×C₇	1	32	yes	—
222	C₂×C₄²	—	2	8	no	—
223	C₂×C₄²	—	2	16	no	—
224	C₂×C₄²	C₇×D₄	1	8	yes	—
225	C₂×C₄²	C₂³×C₇	1	4	yes	—
226	C₂×C₄²	C₂³×C₇	1	4	yes	—
227	C₂×C₄²	C₂³×C₇	1	8	yes	—
228	C₂×C₄²	C₂³×C₇	1	4	yes	—
229	C₂×C₄²	C₂³×C₇	1	4	yes	—
230	C₂×C₄²	C₂³×C₇	1	2	yes	—
231	C₂×C₄²	C₂³×C₇	1	1	yes	characteristic
232	C₂×C₄²	C₂³×C₇	1	1	yes	characteristic
233	C₂×C₄²	C₂³×C₇	1	4	yes	—
234	#95	C₂³×C₇	1	32	yes	—
235	C₂²×C₇	—	2	4	no	—
236	C₂²×C₇	—	2	8	no	—
237	C₂²×C₇	—	4	4	no	—
238	C₂²×C₇	—	4	8	no	—
239	C₂²×C₇	—	4	16	no	—
240	C₂²×C₇	C₂×#124	1	4	yes	—
241	C₂²×C₇	C₂³×D₄	1	1	yes	characteristic
242	C₂²×C₇	C₂²×#41	1	2	yes	—
243	C₂²×C₇	C₂²×#41	1	4	yes	—
244	C₄×C₇	—	2	4	no	—
245	C₄×C₇	—	2	16	no	—
246	C₄×C₇	—	4	4	no	—
247	C₄×C₇	—	4	8	no	—
248	C₄×C₇	C₂×#124	1	4	yes	—
249	C₄×C₇	C₂×#124	1	4	yes	—
250	C₄×C₇	#532	1	8	yes	—
251	C₂⁴	C₂⁴×C₇	1	4	yes	—
252	C₂⁴	C₂⁴×C₇	1	1	yes	characteristic
253	#41	—	4	32	no	—
254	C₂×Q₈	—	2	8	no	—
255	C₂×Q₈	—	2	4	no	—
256	C₂×Q₈	—	2	16	no	—
257	C₂×Q₈	—	2	8	no	—
258	C₂×D₄	—	2	8	no	—
259	C₂×D₄	—	2	16	no	—
260	C₂×D₄	—	2	8	no	—
261	C₂×D₄	—	2	4	no	—
262	C₂×D₄	—	2	16	no	—
263	C₂×D₄	—	2	8	no	—
264	C₂²×C₄	—	2	8	no	—
265	C₂²×C₄	—	2	16	no	—
266	C₂²×C₄	—	2	8	no	—
267	C₂²×C₄	—	2	8	no	—
268	C₂²×C₄	—	2	16	no	—
269	C₂²×C₄	C₂×C₇×D₄	1	8	yes	—
270	C₂²×C₄	C₂×C₇×D₄	1	4	yes	—
271	C₂²×C₄	C₇×#41	1	8	yes	—
272	C₂²×C₄	C₂⁴×C₇	1	4	yes	—
273	C₂²×C₄	C₂⁴×C₇	1	8	yes	—
274	C₂²×C₄	C₂⁴×C₇	1	2	yes	—
275	C₂²×C₄	C₂⁴×C₇	1	8	yes	—
276	C₂²×C₄	C₂⁴×C₇	1	1	yes	characteristic
277	C₂²×C₄	C₂⁴×C₇	1	1	yes	characteristic
278	C₂²×C₄	C₂⁴×C₇	1	2	yes	—
279	C₄⋊C₄	—	2	8	no	—
280	C₄⋊C₄	—	2	8	no	—
281	C₄⋊C₄	—	2	16	no	—
282	C₄⋊C₄	—	2	8	no	—
283	C₂²⋊C₄	—	2	8	no	—
284	C₂²⋊C₄	—	2	16	no	—
285	C₂²⋊C₄	—	2	64	no	—
286	C₄²	—	2	8	no	—
287	C₄²	—	2	16	no	—
288	C₄²	—	2	8	no	—
289	C₄²	—	2	8	no	—
290	C₄²	—	2	8	no	—
291	C₄²	C₂×C₇×D₄	1	8	yes	—
292	C₄²	C₂×C₇×D₄	1	4	yes	—
293	C₄²	C₂×C₇×D₄	1	4	yes	—
294	C₄²	C₇×#41	1	16	yes	complemented
295	C₂×C₇	—	2	4	no	—
296	C₂×C₇	—	8	4	no	—
297	C₂×C₇	C₂²×#124	1	1	yes	characteristic
298	C₂×C₇	C₂×#532	1	2	yes	—
299	C₂×C₇	#3444	1	4	yes	—
300	C₂³	—	2	4	no	—
301	C₂³	—	2	4	no	—
302	C₂³	—	2	16	no	—
303	C₂³	—	2	8	no	—
304	C₂³	C₂²×C₇×D₄	1	2	yes	—
305	C₂³	C₂×C₇×#41	1	4	yes	—
306	C₂³	C₂⁵×C₇	1	1	yes	characteristic
307	Q₈	—	4	8	no	—
308	Q₈	—	4	16	no	—
309	D₄	—	4	16	no	—
310	D₄	—	4	8	no	—
311	D₄	—	4	16	no	—
312	C₂×C₄	—	2	32	no	—
313	C₂×C₄	—	2	4	no	—
314	C₂×C₄	—	2	8	no	—
315	C₂×C₄	—	2	8	no	—
316	C₂×C₄	—	2	16	no	—
317	C₂×C₄	—	2	8	no	—
318	C₂×C₄	—	2	4	no	—
319	C₂×C₄	—	2	16	no	—
320	C₂×C₄	—	2	8	no	—
321	C₂×C₄	—	4	16	no	—
322	C₂×C₄	C₇×#124	1	16	yes	complemented
323	C₂×C₄	C₂²×C₇×D₄	1	4	yes	—
324	C₂×C₄	C₂²×C₇×D₄	1	2	yes	—
325	C₂×C₄	C₂²×C₇×D₄	1	2	yes	—
326	C₂×C₄	C₂²×C₇×D₄	1	4	yes	—
327	C₂×C₄	C₂×C₇×#41	1	16	yes	—
328	C₂×C₄	C₂×C₇×#41	1	4	yes	—
329	C₂×C₄	C₂×C₇×#41	1	4	yes	—
330	C₂×C₄	C₂×C₇×#41	1	8	yes	—
331	C₇	#36639	1	1	yes	characteristic, direct product factor, Sylow
332	C₂²	—	2	4	no	—
333	C₂²	—	2	8	no	—
334	C₂²	—	4	4	no	—
335	C₂²	—	4	8	no	—
336	C₂²	—	4	16	no	—
337	C₂²	C₂×C₇×#124	1	4	yes	—
338	C₂²	C₂³×C₇×D₄	1	1	yes	characteristic
339	C₂²	C₂²×C₇×#41	1	2	yes	—
340	C₂²	C₂²×C₇×#41	1	4	yes	—
341	C₄	—	2	4	no	—
342	C₄	—	2	16	no	—
343	C₄	—	4	4	no	—
344	C₄	—	4	8	no	—
345	C₄	C₂×C₇×#124	1	4	yes	—
346	C₄	C₂×C₇×#124	1	4	yes	—
347	C₄	C₇×#532	1	8	yes	—
348	C₂	—	2	4	no	—
349	C₂	—	8	4	no	—
350	C₂	C₂²×C₇×#124	1	1	yes	characteristic
351	C₂	C₂×C₇×#532	1	2	yes	—
352	C₂	C₇×#3444	1	4	yes	—

	subgroup	G/H
center	C₂²×C₄×C₇	C₂⁴
commutator, Frattini, cosocle	C₂³	C₂⁵×C₇
socle	C₂³×C₇	C₂⁵
prime radical	C₂×C₇×#532	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂²×C₄×C₇ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×C₇×#533	C₂	semidirect
3. C₇×#2862	C₂	semidirect
4. C₇×#2861	C₂	semidirect
5. C₇×#2858	C₂	semidirect
6. C₇×#2850	C₂	semidirect
7. C₄×C₇×#124	C₂	semidirect
8. C₇×#2297	C₂	semidirect
9. C₂×C₄³×C₇	C₂	semidirect
38. #36639	C₇	direct (N × H)
81. C₂×#533	C₂×C₇	semidirect
83. #2862	C₂×C₇	semidirect
84. #2861	C₂×C₇	semidirect
85. #2858	C₂×C₇	semidirect
86. #2850	C₂×C₇	semidirect
87. C₄×#124	C₂×C₇	semidirect
88. #2297	C₂×C₇	semidirect
89. C₂×C₄³	C₂×C₇	semidirect
133. C₄²×C₇	#41	semidirect
184. C₂×C₄×C₇	#124	semidirect
294. C₄²	C₇×#41	semidirect
322. C₂×C₄	C₇×#124	semidirect
331. C₇	#36639	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₇×#533	2
C₂×C₇×#532	2
C₇×#2862	2
C₇×#2861	2
C₇×#2858	2
C₇×#2850	2
C₄×C₇×#124	2
C₇×#2297	2
C₂×C₄³×C₇	2
#36639	7

Subgroups — C7×#36639 #13213855

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×#36639 #13213855