Subgroups — C₇×#23816 #13201032

Order 1792.

778 conjugacy classes of subgroups.

The subgroup lattice has 350 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×C₇×C₄⋊Q₈	C₂	1	1	yes	maximal, characteristic
2	C₄×C₇×C₄⋊Q₈	C₂	1	1	yes	maximal, characteristic
3	C₇×#2100	C₂	1	2	yes	maximal
4	C₇×#2072	C₂	1	2	yes	maximal
5	C₇×#2043	C₂	1	2	yes	maximal
6	C₇×#2024	C₂	1	2	yes	maximal
7	C₄×C₇×#103	C₂	1	4	yes	maximal
8	C₇×#1744	C₂	1	1	yes	maximal, characteristic
9	C₂×C₇×C₄⋊Q₈	C₂²	1	1	yes	characteristic
10	C₂×C₇×C₄⋊Q₈	C₂²	1	1	yes	characteristic
11	C₂×C₄×C₇×Q₈	—	2	1	no	—
12	C₂×C₄×C₇×Q₈	C₂²	1	2	yes	—
13	C₂×C₇×#103	C₂²	1	4	yes	—
14	C₂×C₇×#103	C₂²	1	4	yes	—
15	C₂×C₄×C₇×C₈	C₂²	1	2	yes	—
16	C₇×#391	—	2	1	no	—
17	C₇×#391	C₂²	1	2	yes	—
18	C₇×#389	—	2	1	no	—
19	C₇×#389	—	2	1	no	—
20	C₇×#389	—	2	1	no	—
21	C₇×#389	C₂²	1	2	yes	—
22	C₇×#384	C₂²	1	1	yes	characteristic
23	C₇×#384	C₂²	1	1	yes	characteristic
24	C₄×C₇×C₄⋊C₄	—	2	1	no	—
25	C₄×C₇×C₄⋊C₄	—	2	1	no	—
26	C₄×C₇×C₄⋊C₄	—	2	1	no	—
27	C₄×C₇×C₄⋊C₄	C₂²	1	2	yes	—
28	C₄³×C₇	C₂²	1	1	yes	characteristic
29	C₇×#340	C₂²	1	4	yes	—
30	C₇×#340	C₂²	1	4	yes	—
31	C₇×#336	C₂²	1	2	yes	—
32	C₇×#336	C₂²	1	2	yes	—
33	#23816	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
34	C₂²×C₇×Q₈	—	2	1	no	—
35	C₂²×C₇×Q₈	C₂³	1	2	yes	—
36	C₂²×C₇×C₈	C₂³	1	4	yes	—
37	C₇×C₄⋊Q₈	—	2	4	no	—
38	C₇×C₄⋊Q₈	—	2	4	no	—
39	C₄×C₇×Q₈	—	2	8	no	—
40	C₄×C₇×Q₈	—	4	4	no	—
41	C₂×C₇×C₄⋊C₄	—	2	1	no	—
42	C₂×C₇×C₄⋊C₄	—	2	1	no	—
43	C₂×C₇×C₄⋊C₄	—	2	1	no	—
44	C₂×C₇×C₄⋊C₄	—	2	1	no	—
45	C₂×C₇×C₄⋊C₄	—	2	1	no	—
46	C₂×C₇×C₄⋊C₄	—	2	1	no	—
47	C₂×C₇×C₄⋊C₄	—	2	1	no	—
48	C₂×C₇×C₄⋊C₄	—	2	2	no	—
49	C₂×C₇×C₄⋊C₄	—	2	2	no	—
50	C₂×C₇×C₄⋊C₄	C₂³	1	2	yes	—
51	C₂×C₇×C₄⋊C₄	C₂³	1	2	yes	—
52	C₂×C₇×C₄⋊C₄	C₂³	1	2	yes	—
53	C₂×C₄²×C₇	—	2	2	no	—
54	C₂×C₄²×C₇	—	2	2	no	—
55	C₂×C₄²×C₇	D₄	1	1	yes	characteristic
56	C₂×C₄²×C₇	D₄	1	1	yes	characteristic
57	C₂×C₄²×C₇	D₄	1	1	yes	characteristic
58	C₂×C₄²×C₇	D₄	1	1	yes	characteristic
59	C₂×C₄²×C₇	C₂³	1	1	yes	characteristic
60	C₂×C₄²×C₇	C₂³	1	1	yes	characteristic
61	C₂×C₄²×C₇	C₂³	1	1	yes	characteristic
62	C₇×#103	—	2	8	no	—
63	C₇×#103	—	2	8	no	—
64	C₄×C₇×C₈	—	2	4	no	—
65	C₇×#95	—	2	2	no	—
66	C₇×#95	—	2	2	no	—
67	C₇×#95	—	2	2	no	—
68	C₇×#95	—	2	2	no	—
69	C₄×C₄⋊Q₈	C₂×C₇	1	1	yes	characteristic
70	C₄×C₄⋊Q₈	C₂×C₇	1	1	yes	characteristic
71	#2100	C₂×C₇	1	2	yes	—
72	#2072	C₂×C₇	1	2	yes	—
73	#2043	C₂×C₇	1	2	yes	—
74	#2024	C₂×C₇	1	2	yes	—
75	C₄×#103	C₂×C₇	1	4	yes	—
76	#1744	C₂×C₇	1	1	yes	characteristic
77	C₂×C₇×Q₈	—	2	4	no	—
78	C₂×C₇×Q₈	—	2	2	no	—
79	C₂×C₇×Q₈	—	2	4	no	—
80	C₂×C₇×Q₈	—	2	4	no	—
81	C₂×C₇×Q₈	—	4	2	no	—
82	C₂×C₇×Q₈	—	4	2	no	—
83	C₂×C₇×Q₈	#41	1	4	yes	—
84	C₂²×C₄×C₇	—	2	2	no	—
85	C₂²×C₄×C₇	—	2	2	no	—
86	C₂²×C₄×C₇	—	2	2	no	—
87	C₂²×C₄×C₇	—	2	2	no	—
88	C₂²×C₄×C₇	C₂×D₄	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×C₄×C₇	C₂×D₄	1	1	yes	characteristic
92	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
93	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
94	C₂²×C₄×C₇	C₂⁴	1	1	yes	characteristic
95	C₂×C₇×C₈	—	2	4	no	—
96	C₂×C₇×C₈	—	2	4	no	—
97	C₂×C₇×C₈	—	2	4	no	—
98	C₇×C₄⋊C₄	—	2	4	no	—
99	C₇×C₄⋊C₄	—	2	4	no	—
100	C₇×C₄⋊C₄	—	2	4	no	—
101	C₇×C₄⋊C₄	—	2	2	no	—
102	C₇×C₄⋊C₄	—	2	4	no	—
103	C₇×C₄⋊C₄	—	2	4	no	—
104	C₇×C₄⋊C₄	—	4	2	no	—
105	C₇×C₄⋊C₄	—	4	2	no	—
106	C₇×C₄⋊C₄	—	4	2	no	—
107	C₇×C₄⋊C₄	—	4	4	no	—
108	C₇×C₄⋊C₄	—	4	4	no	—
109	C₇×C₄⋊C₄	—	4	2	no	—
110	C₇×C₄⋊C₄	#41	1	4	yes	—
111	C₄²×C₇	—	2	2	no	—
112	C₄²×C₇	—	2	2	no	—
113	C₄²×C₇	—	4	1	no	—
114	C₄²×C₇	—	4	1	no	—
115	C₄²×C₇	—	4	4	no	—
116	C₄²×C₇	—	4	4	no	—
117	C₄²×C₇	SD₁₆	1	4	yes	—
118	C₄²×C₇	Q₁₆	1	4	yes	—
119	C₄²×C₇	#41	1	4	yes	—
120	C₂×C₄⋊Q₈	C₂²×C₇	1	1	yes	characteristic
121	C₂×C₄⋊Q₈	C₂²×C₇	1	1	yes	characteristic
122	C₂×C₄×Q₈	—	2	1	no	—
123	C₂×C₄×Q₈	C₂²×C₇	1	2	yes	—
124	C₂×#103	C₂²×C₇	1	4	yes	—
125	C₂×#103	C₂²×C₇	1	4	yes	—
126	C₂×C₄×C₈	C₂²×C₇	1	2	yes	—
127	#391	—	2	1	no	—
128	#391	C₂²×C₇	1	2	yes	—
129	#389	—	2	1	no	—
130	#389	—	2	1	no	—
131	#389	—	2	1	no	—
132	#389	C₂²×C₇	1	2	yes	—
133	#384	C₂²×C₇	1	1	yes	characteristic
134	#384	C₂²×C₇	1	1	yes	characteristic
135	C₄×C₄⋊C₄	—	2	1	no	—
136	C₄×C₄⋊C₄	—	2	1	no	—
137	C₄×C₄⋊C₄	—	2	1	no	—
138	C₄×C₄⋊C₄	C₂²×C₇	1	2	yes	—
139	C₄³	C₂²×C₇	1	1	yes	characteristic
140	#340	C₂²×C₇	1	4	yes	—
141	#340	C₂²×C₇	1	4	yes	—
142	#336	C₂²×C₇	1	2	yes	—
143	#336	C₂²×C₇	1	2	yes	—
144	C₂³×C₇	C₂²×D₄	1	1	yes	characteristic
145	C₇×Q₈	—	2	8	no	—
146	C₇×Q₈	—	4	8	no	—
147	C₇×Q₈	—	4	4	no	—
148	C₂×C₄×C₇	—	2	1	no	—
149	C₂×C₄×C₇	—	2	1	no	—
150	C₂×C₄×C₇	—	2	1	no	—
151	C₂×C₄×C₇	—	2	1	no	—
152	C₂×C₄×C₇	—	2	1	no	—
153	C₂×C₄×C₇	—	2	1	no	—
154	C₂×C₄×C₇	—	2	1	no	—
155	C₂×C₄×C₇	—	2	1	no	—
156	C₂×C₄×C₇	—	2	1	no	—
157	C₂×C₄×C₇	—	2	1	no	—
158	C₂×C₄×C₇	—	2	4	no	—
159	C₂×C₄×C₇	—	2	4	no	—
160	C₂×C₄×C₇	—	4	2	no	—
161	C₂×C₄×C₇	—	4	2	no	—
162	C₂×C₄×C₇	—	4	2	no	—
163	C₂×C₄×C₇	—	4	2	no	—
164	C₂×C₄×C₇	—	4	2	no	—
165	C₂×C₄×C₇	—	4	2	no	—
166	C₂×C₄×C₇	—	4	2	no	—
167	C₂×C₄×C₇	—	4	2	no	—
168	C₂×C₄×C₇	—	4	2	no	—
169	C₂×C₄×C₇	—	4	2	no	—
170	C₂×C₄×C₇	#124	1	2	yes	—
171	C₂×C₄×C₇	#124	1	2	yes	—
172	C₂×C₄×C₇	C₂×SD₁₆	1	2	yes	—
173	C₂×C₄×C₇	C₂×SD₁₆	1	2	yes	—
174	C₂×C₄×C₇	C₂×SD₁₆	1	2	yes	—
175	C₂×C₄×C₇	C₂×Q₁₆	1	2	yes	—
176	C₂×C₄×C₇	C₂×Q₁₆	1	2	yes	—
177	C₂×C₄×C₇	C₂×Q₁₆	1	2	yes	—
178	C₂×C₄×C₇	C₂×#41	1	2	yes	—
179	C₂×C₄×C₇	C₂×#41	1	2	yes	—
180	C₂×C₄×C₇	C₂×#41	1	2	yes	—
181	C₇×C₈	—	4	4	no	—
182	C₂²×Q₈	—	2	1	no	—
183	C₂²×Q₈	C₂³×C₇	1	2	yes	—
184	C₂²×C₈	C₂³×C₇	1	4	yes	—
185	C₄⋊Q₈	—	2	4	no	—
186	C₄⋊Q₈	—	2	4	no	—
187	C₄×Q₈	—	2	8	no	—
188	C₄×Q₈	—	4	4	no	—
189	C₂×C₄⋊C₄	—	2	1	no	—
190	C₂×C₄⋊C₄	—	2	1	no	—
191	C₂×C₄⋊C₄	—	2	1	no	—
192	C₂×C₄⋊C₄	—	2	1	no	—
193	C₂×C₄⋊C₄	—	2	2	no	—
194	C₂×C₄⋊C₄	—	2	2	no	—
195	C₂×C₄⋊C₄	—	2	1	no	—
196	C₂×C₄⋊C₄	—	2	1	no	—
197	C₂×C₄⋊C₄	—	2	1	no	—
198	C₂×C₄⋊C₄	C₂³×C₇	1	2	yes	—
199	C₂×C₄⋊C₄	C₂³×C₇	1	2	yes	—
200	C₂×C₄⋊C₄	C₂³×C₇	1	2	yes	—
201	C₂×C₄²	—	2	2	no	—
202	C₂×C₄²	—	2	2	no	—
203	C₂×C₄²	C₇×D₄	1	1	yes	characteristic
204	C₂×C₄²	C₇×D₄	1	1	yes	characteristic
205	C₂×C₄²	C₇×D₄	1	1	yes	characteristic
206	C₂×C₄²	C₇×D₄	1	1	yes	characteristic
207	C₂×C₄²	C₂³×C₇	1	1	yes	characteristic
208	C₂×C₄²	C₂³×C₇	1	1	yes	characteristic
209	C₂×C₄²	C₂³×C₇	1	1	yes	characteristic
210	#103	—	2	8	no	—
211	#103	—	2	8	no	—
212	C₄×C₈	—	2	4	no	—
213	#95	—	2	2	no	—
214	#95	—	2	2	no	—
215	#95	—	2	2	no	—
216	#95	—	2	2	no	—
217	C₂²×C₇	#525	1	1	yes	characteristic
218	C₂²×C₇	#525	1	1	yes	characteristic
219	C₂²×C₇	C₂×#124	1	1	yes	characteristic
220	C₂²×C₇	C₂²×SD₁₆	1	1	yes	characteristic
221	C₂²×C₇	C₂²×Q₁₆	1	1	yes	characteristic
222	C₂²×C₇	#575	1	1	yes	characteristic
223	C₂²×C₇	#575	1	1	yes	characteristic
224	C₄×C₇	—	2	2	no	—
225	C₄×C₇	—	2	2	no	—
226	C₄×C₇	—	2	2	no	—
227	C₄×C₇	—	2	2	no	—
228	C₄×C₇	—	2	2	no	—
229	C₄×C₇	—	4	4	no	—
230	C₄×C₇	—	4	4	no	—
231	C₄×C₇	—	8	2	no	—
232	C₄×C₇	—	8	2	no	—
233	C₄×C₇	#487	1	4	yes	—
234	C₄×C₇	#529	1	4	yes	—
235	C₂×Q₈	—	2	4	no	—
236	C₂×Q₈	—	2	2	no	—
237	C₂×Q₈	—	2	4	no	—
238	C₂×Q₈	—	2	4	no	—
239	C₂×Q₈	—	4	2	no	—
240	C₂×Q₈	—	4	2	no	—
241	C₂×Q₈	C₇×#41	1	4	yes	—
242	C₂²×C₄	—	2	2	no	—
243	C₂²×C₄	—	2	2	no	—
244	C₂²×C₄	—	2	2	no	—
245	C₂²×C₄	—	2	2	no	—
246	C₂²×C₄	C₂×C₇×D₄	1	1	yes	characteristic
247	C₂²×C₄	C₂×C₇×D₄	1	1	yes	characteristic
248	C₂²×C₄	C₂×C₇×D₄	1	1	yes	characteristic
249	C₂²×C₄	C₂×C₇×D₄	1	1	yes	characteristic
250	C₂²×C₄	C₂×C₇×D₄	1	1	yes	characteristic
251	C₂²×C₄	C₂×C₇×D₄	1	1	yes	characteristic
252	C₂²×C₄	C₂⁴×C₇	1	1	yes	characteristic
253	C₂×C₈	—	2	4	no	—
254	C₂×C₈	—	2	4	no	—
255	C₂×C₈	—	2	4	no	—
256	C₄⋊C₄	—	2	4	no	—
257	C₄⋊C₄	—	2	4	no	—
258	C₄⋊C₄	—	2	4	no	—
259	C₄⋊C₄	—	2	2	no	—
260	C₄⋊C₄	—	2	4	no	—
261	C₄⋊C₄	—	2	4	no	—
262	C₄⋊C₄	—	4	2	no	—
263	C₄⋊C₄	—	4	2	no	—
264	C₄⋊C₄	—	4	2	no	—
265	C₄⋊C₄	—	4	4	no	—
266	C₄⋊C₄	—	4	4	no	—
267	C₄⋊C₄	—	4	2	no	—
268	C₄⋊C₄	C₇×#41	1	4	yes	—
269	C₄²	—	2	2	no	—
270	C₄²	—	2	2	no	—
271	C₄²	—	4	1	no	—
272	C₄²	—	4	1	no	—
273	C₄²	—	4	4	no	—
274	C₄²	—	4	4	no	—
275	C₄²	C₇×SD₁₆	1	4	yes	—
276	C₄²	C₇×Q₁₆	1	4	yes	—
277	C₄²	C₇×#41	1	4	yes	—
278	C₂×C₇	#2398	1	1	yes	characteristic
279	C₂×C₇	#3101	1	1	yes	characteristic
280	C₂×C₇	#3103	1	1	yes	characteristic
281	C₂×C₇	#3104	1	1	yes	characteristic
282	C₂×C₇	#3104	1	1	yes	characteristic
283	C₂×C₇	C₂×#487	1	1	yes	characteristic
284	C₂×C₇	#3135	1	1	yes	characteristic
285	C₂³	C₂²×C₇×D₄	1	1	yes	characteristic
286	Q₈	—	2	8	no	—
287	Q₈	—	4	8	no	—
288	Q₈	—	4	4	no	—
289	C₂×C₄	—	2	1	no	—
290	C₂×C₄	—	2	1	no	—
291	C₂×C₄	—	2	1	no	—
292	C₂×C₄	—	2	1	no	—
293	C₂×C₄	—	2	1	no	—
294	C₂×C₄	—	2	1	no	—
295	C₂×C₄	—	2	1	no	—
296	C₂×C₄	—	2	1	no	—
297	C₂×C₄	—	2	1	no	—
298	C₂×C₄	—	2	1	no	—
299	C₂×C₄	—	2	4	no	—
300	C₂×C₄	—	2	4	no	—
301	C₂×C₄	—	4	2	no	—
302	C₂×C₄	—	4	2	no	—
303	C₂×C₄	—	4	2	no	—
304	C₂×C₄	—	4	2	no	—
305	C₂×C₄	—	4	2	no	—
306	C₂×C₄	—	4	2	no	—
307	C₂×C₄	—	4	2	no	—
308	C₂×C₄	—	4	2	no	—
309	C₂×C₄	—	4	2	no	—
310	C₂×C₄	—	4	2	no	—
311	C₂×C₄	C₇×#124	1	2	yes	—
312	C₂×C₄	C₇×#124	1	2	yes	—
313	C₂×C₄	C₂×C₇×SD₁₆	1	2	yes	—
314	C₂×C₄	C₂×C₇×SD₁₆	1	2	yes	—
315	C₂×C₄	C₂×C₇×SD₁₆	1	2	yes	—
316	C₂×C₄	C₂×C₇×Q₁₆	1	2	yes	—
317	C₂×C₄	C₂×C₇×Q₁₆	1	2	yes	—
318	C₂×C₄	C₂×C₇×Q₁₆	1	2	yes	—
319	C₂×C₄	C₂×C₇×#41	1	2	yes	—
320	C₂×C₄	C₂×C₇×#41	1	2	yes	—
321	C₂×C₄	C₂×C₇×#41	1	2	yes	—
322	C₈	—	4	4	no	—
323	C₇	#23816	1	1	yes	characteristic, direct product factor, Sylow
324	C₂²	C₇×#525	1	1	yes	characteristic
325	C₂²	C₇×#525	1	1	yes	characteristic
326	C₂²	C₂×C₇×#124	1	1	yes	characteristic
327	C₂²	C₂²×C₇×SD₁₆	1	1	yes	characteristic
328	C₂²	C₂²×C₇×Q₁₆	1	1	yes	characteristic
329	C₂²	C₇×#575	1	1	yes	characteristic
330	C₂²	C₇×#575	1	1	yes	characteristic
331	C₄	—	2	2	no	—
332	C₄	—	2	2	no	—
333	C₄	—	2	2	no	—
334	C₄	—	2	2	no	—
335	C₄	—	2	2	no	—
336	C₄	—	4	4	no	—
337	C₄	—	4	4	no	—
338	C₄	—	8	2	no	—
339	C₄	—	8	2	no	—
340	C₄	C₇×#487	1	4	yes	—
341	C₄	C₇×#529	1	4	yes	—
342	C₂	C₇×#2398	1	1	yes	characteristic
343	C₂	C₇×#3101	1	1	yes	characteristic
344	C₂	C₇×#3103	1	1	yes	characteristic
345	C₂	C₇×#3104	1	1	yes	characteristic
346	C₂	C₇×#3104	1	1	yes	characteristic
347	C₂	C₂×C₇×#487	1	1	yes	characteristic
348	C₂	C₇×#3135	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄×C₇	C₂×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₇
socle, prime radical	C₂³×C₇	C₂²×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄×C₇ ◁ C₄³×C₇ ◁ G

Complements

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

N	H ≅ G/N	type
33. #23816	C₇	direct (N × H)
323. C₇	#23816	direct (N × H)

Maximal subgroups

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

subgroup	index
C₄×C₇×C₄⋊Q₈	2
C₇×#2100	2
C₇×#2072	2
C₇×#2043	2
C₇×#2024	2
C₄×C₇×#103	2
C₇×#1744	2
#23816	7

Subgroups — C7×#23816 #13201032

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×#23816 #13201032