Subgroups — C₇×#23720 #13200936

Order 1792.

744 conjugacy classes of subgroups.

The subgroup lattice has 384 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×C₇×#137	C₂	1	2	yes	maximal
2	C₂×C₇×#441	C₂	1	1	yes	maximal, characteristic, complemented
3	C₇×#2923	C₂	1	1	yes	maximal, characteristic
4	C₇×#2090	C₂	1	1	yes	maximal, characteristic, complemented
5	C₇×#2079	C₂	1	1	yes	maximal, characteristic
6	C₇×#2064	C₂	1	2	yes	maximal, complemented
7	C₇×#2052	C₂	1	2	yes	maximal
8	C₇×#2047	C₂	1	1	yes	maximal, characteristic, complemented
9	C₇×#2020	C₂	1	1	yes	maximal, characteristic
10	C₇×#1778	C₂	1	1	yes	maximal, characteristic, complemented
11	C₄×C₇×#101	C₂	1	2	yes	maximal, complemented
12	C₂×C₇×#137	C₂²	1	2	yes	—
13	C₂²×C₇×Q₁₆	C₂²	1	1	yes	characteristic
14	C₄×C₇×#41	C₂²	1	1	yes	characteristic
15	C₂×C₄×C₇×Q₈	C₂²	1	2	yes	—
16	C₇×#441	—	2	2	no	—
17	C₇×#441	—	2	2	no	—
18	C₇×#441	—	2	2	no	—
19	C₇×#440	—	2	2	no	—
20	C₄×C₇×Q₁₆	—	2	2	no	—
21	C₄×C₇×SD₁₆	—	2	2	no	—
22	C₇×#433	—	2	2	no	—
23	C₇×#432	—	2	2	no	—
24	C₂×C₇×#108	C₂²	1	2	yes	—
25	C₇×#428	C₂²	1	2	yes	—
26	C₂×C₇×#106	C₂²	1	1	yes	characteristic
27	C₇×#424	C₂²	1	1	yes	characteristic
28	C₇×#424	C₂²	1	1	yes	characteristic
29	C₇×#419	C₂²	1	2	yes	—
30	C₇×#417	C₂²	1	2	yes	—
31	C₂×C₇×#103	C₂²	1	2	yes	—
32	C₂×C₇×#101	C₂²	1	2	yes	—
33	C₂×C₇×#101	C₂²	1	2	yes	—
34	C₇×#408	C₂²	1	1	yes	characteristic
35	C₇×#408	C₂²	1	1	yes	characteristic
36	C₄×C₇×M₁₆	C₂²	1	1	yes	characteristic
37	C₄×C₇×M₁₆	C₂²	1	2	yes	—
38	C₂×C₇×#97	C₂²	1	1	yes	characteristic
39	C₇×#343	C₂²	1	2	yes	—
40	C₇×#343	C₂²	1	2	yes	—
41	C₇×#341	C₂²	1	1	yes	characteristic
42	C₇×#341	C₂²	1	1	yes	characteristic
43	#23720	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
44	C₂×C₇×#41	C₂³	1	1	yes	characteristic
45	C₂²×C₇×Q₈	C₂³	1	2	yes	—
46	C₇×#137	—	2	4	no	—
47	C₇×#137	—	2	4	no	—
48	C₂×C₇×Q₁₆	—	2	2	no	—
49	C₂×C₇×Q₁₆	—	2	4	no	—
50	C₂×C₇×Q₁₆	—	2	2	no	—
51	C₂×C₇×SD₁₆	—	2	2	no	—
52	C₂×C₇×M₁₆	C₂³	1	2	yes	—
53	C₂×C₇×M₁₆	C₂³	1	1	yes	characteristic
54	C₂×C₇×M₁₆	C₂³	1	1	yes	characteristic
55	C₂×C₇×M₁₆	C₂³	1	2	yes	—
56	C₂²×C₇×C₈	C₂³	1	1	yes	characteristic
57	C₂²×C₇×C₈	C₂³	1	1	yes	characteristic
58	C₄×C₇×Q₈	—	2	2	no	—
59	C₄×C₇×Q₈	—	2	2	no	—
60	C₄×C₇×Q₈	—	2	2	no	—
61	C₄×C₇×Q₈	—	2	2	no	—
62	C₄×C₇×Q₈	D₄	1	2	yes	—
63	C₄×C₇×D₄	—	2	2	no	—
64	C₄×C₇×D₄	D₄	1	2	yes	—
65	C₇×#117	—	2	1	no	—
66	C₇×#117	C₂³	1	1	yes	characteristic
67	C₂×C₇×C₄⋊C₄	—	2	2	no	—
68	C₂×C₇×C₄⋊C₄	C₂³	1	2	yes	—
69	C₂×C₄²×C₇	—	2	1	no	—
70	C₂×C₄²×C₇	—	2	2	no	—
71	C₂×C₄²×C₇	C₂³	1	1	yes	characteristic
72	C₇×#108	—	2	4	no	—
73	C₇×#107	—	2	2	no	—
74	C₇×#106	—	2	2	no	—
75	C₇×#106	—	2	2	no	—
76	C₇×C₄⋊C₈	—	2	2	no	—
77	C₇×C₄⋊C₈	—	2	2	no	—
78	C₇×#103	—	2	2	no	—
79	C₇×#103	—	2	4	no	—
80	C₇×#103	—	2	2	no	—
81	C₇×#103	—	2	2	no	—
82	C₇×#102	—	2	2	no	—
83	C₇×#101	—	2	4	no	—
84	C₇×#101	—	2	4	no	—
85	C₇×C₂²⋊C₈	—	2	2	no	—
86	C₇×C₂²⋊C₈	—	2	2	no	—
87	C₇×#97	—	2	1	no	—
88	C₇×#97	—	2	2	no	—
89	C₇×#97	D₄	1	2	yes	—
90	C₇×#97	D₄	1	2	yes	—
91	C₄×C₇×C₈	—	2	1	no	—
92	C₄×C₇×C₈	—	2	2	no	—
93	C₄×#137	C₂×C₇	1	2	yes	—
94	C₂×#441	C₂×C₇	1	1	yes	characteristic, complemented
95	#2923	C₂×C₇	1	1	yes	characteristic
96	#2090	C₂×C₇	1	1	yes	characteristic, complemented
97	#2079	C₂×C₇	1	1	yes	characteristic
98	#2064	C₂×C₇	1	2	yes	complemented
99	#2052	C₂×C₇	1	2	yes	—
100	#2047	C₂×C₇	1	1	yes	characteristic, complemented
101	#2020	C₂×C₇	1	1	yes	characteristic
102	#1778	C₂×C₇	1	1	yes	characteristic, complemented
103	C₄×#101	C₂×C₇	1	2	yes	complemented
104	C₇×#41	—	2	2	no	—
105	C₇×#41	—	2	2	no	—
106	C₂×C₇×Q₈	—	2	2	no	—
107	C₂×C₇×Q₈	—	2	2	no	—
108	C₂×C₇×Q₈	—	2	2	no	—
109	C₂×C₇×Q₈	—	2	2	no	—
110	C₂×C₇×Q₈	—	2	2	no	—
111	C₂×C₇×Q₈	C₂×D₄	1	1	yes	characteristic
112	C₂×C₇×Q₈	#41	1	4	yes	—
113	C₂×C₇×D₄	—	2	1	no	—
114	C₂×C₇×D₄	C₂×D₄	1	1	yes	characteristic
115	C₂²×C₄×C₇	—	2	2	no	—
116	C₂²×C₄×C₇	—	2	1	no	—
117	C₂²×C₄×C₇	—	2	2	no	—
118	C₂²×C₄×C₇	—	2	2	no	—
119	C₂²×C₄×C₇	#41	1	2	yes	—
120	C₂²×C₄×C₇	C₂⁴	1	1	yes	characteristic
121	C₇×Q₁₆	—	2	8	no	—
122	C₇×Q₁₆	—	2	8	no	—
123	C₇×SD₁₆	—	2	8	no	—
124	C₇×M₁₆	—	2	2	no	—
125	C₇×M₁₆	—	2	2	no	—
126	C₇×M₁₆	—	2	2	no	—
127	C₇×M₁₆	—	2	2	no	—
128	C₇×M₁₆	—	2	2	no	—
129	C₇×M₁₆	#41	1	4	yes	—
130	C₂×C₇×C₈	—	2	2	no	—
131	C₂×C₇×C₈	—	2	2	no	—
132	C₂×C₇×C₈	—	2	2	no	—
133	C₂×C₇×C₈	—	2	1	no	—
134	C₂×C₇×C₈	—	2	1	no	—
135	C₂×C₇×C₈	—	2	2	no	—
136	C₂×C₇×C₈	C₂×D₄	1	2	yes	—
137	C₂×C₇×C₈	C₂×D₄	1	2	yes	—
138	C₇×C₄⋊C₄	—	2	2	no	—
139	C₇×C₄⋊C₄	—	2	2	no	—
140	C₇×C₄⋊C₄	—	2	2	no	—
141	C₇×C₄⋊C₄	—	4	2	no	—
142	C₇×C₄⋊C₄	—	4	2	no	—
143	C₇×C₄⋊C₄	C₂×D₄	1	2	yes	—
144	C₇×C₂²⋊C₄	—	2	1	no	—
145	C₇×C₂²⋊C₄	—	2	2	no	—
146	C₄²×C₇	—	2	2	no	—
147	C₄²×C₇	—	2	1	no	—
148	C₄²×C₇	—	4	2	no	—
149	C₄²×C₇	—	4	2	no	—
150	C₄²×C₇	C₂×D₄	1	2	yes	—
151	C₄²×C₇	C₂×D₄	1	1	yes	characteristic
152	C₄²×C₇	C₂×D₄	1	1	yes	characteristic
153	C₂×#137	C₂²×C₇	1	2	yes	—
154	C₂²×Q₁₆	C₂²×C₇	1	1	yes	characteristic
155	C₄×#41	C₂²×C₇	1	1	yes	characteristic
156	C₂×C₄×Q₈	C₂²×C₇	1	2	yes	—
157	#441	—	2	2	no	—
158	#441	—	2	2	no	—
159	#441	—	2	2	no	—
160	#440	—	2	2	no	—
161	C₄×Q₁₆	—	2	2	no	—
162	C₄×SD₁₆	—	2	2	no	—
163	#433	—	2	2	no	—
164	#432	—	2	2	no	—
165	C₂×#108	C₂²×C₇	1	2	yes	—
166	#428	C₂²×C₇	1	2	yes	—
167	C₂×#106	C₂²×C₇	1	1	yes	characteristic
168	#424	C₂²×C₇	1	1	yes	characteristic
169	#424	C₂²×C₇	1	1	yes	characteristic
170	#419	C₂²×C₇	1	2	yes	—
171	#417	C₂²×C₇	1	2	yes	—
172	C₂×#103	C₂²×C₇	1	2	yes	—
173	C₂×#101	C₂²×C₇	1	2	yes	—
174	C₂×#101	C₂²×C₇	1	2	yes	—
175	#408	C₂²×C₇	1	1	yes	characteristic
176	#408	C₂²×C₇	1	1	yes	characteristic
177	C₄×M₁₆	C₂²×C₇	1	1	yes	characteristic
178	C₄×M₁₆	C₂²×C₇	1	2	yes	—
179	C₂×#97	C₂²×C₇	1	1	yes	characteristic
180	#343	C₂²×C₇	1	2	yes	—
181	#343	C₂²×C₇	1	2	yes	—
182	#341	C₂²×C₇	1	1	yes	characteristic
183	#341	C₂²×C₇	1	1	yes	characteristic
184	C₂³×C₇	—	2	1	no	—
185	C₂³×C₇	C₂×#41	1	1	yes	characteristic
186	C₇×Q₈	—	2	4	no	—
187	C₇×Q₈	—	2	4	no	—
188	C₇×Q₈	—	2	2	no	—
189	C₇×Q₈	—	4	2	no	—
190	C₇×Q₈	—	4	2	no	—
191	C₇×D₄	—	2	2	no	—
192	C₇×D₄	—	4	2	no	—
193	C₂×C₄×C₇	—	2	2	no	—
194	C₂×C₄×C₇	—	2	1	no	—
195	C₂×C₄×C₇	—	2	4	no	—
196	C₂×C₄×C₇	—	4	2	no	—
197	C₂×C₄×C₇	—	4	2	no	—
198	C₂×C₄×C₇	—	4	2	no	—
199	C₂×C₄×C₇	—	4	1	no	—
200	C₂×C₄×C₇	—	4	1	no	—
201	C₂×C₄×C₇	—	4	2	no	—
202	C₂×C₄×C₇	—	8	2	no	—
203	C₂×C₄×C₇	—	8	2	no	—
204	C₂×C₄×C₇	#121	1	2	yes	—
205	C₂×C₄×C₇	#121	1	2	yes	—
206	C₂×C₄×C₇	C₂²×D₄	1	1	yes	characteristic
207	C₂×C₄×C₇	C₂²×D₄	1	1	yes	characteristic
208	C₂×C₄×C₇	C₂×#41	1	2	yes	—
209	C₂×C₄×C₇	C₂×#41	1	2	yes	—
210	C₇×C₈	—	2	2	no	—
211	C₇×C₈	—	2	4	no	—
212	C₇×C₈	—	2	2	no	—
213	C₇×C₈	—	4	1	no	—
214	C₇×C₈	—	4	2	no	—
215	C₇×C₈	—	4	1	no	—
216	C₂×#41	C₂³×C₇	1	1	yes	characteristic
217	C₂²×Q₈	C₂³×C₇	1	2	yes	—
218	#137	—	2	4	no	—
219	#137	—	2	4	no	—
220	C₂×Q₁₆	—	2	2	no	—
221	C₂×Q₁₆	—	2	4	no	—
222	C₂×Q₁₆	—	2	2	no	—
223	C₂×SD₁₆	—	2	2	no	—
224	C₂×M₁₆	C₂³×C₇	1	2	yes	—
225	C₂×M₁₆	C₂³×C₇	1	1	yes	characteristic
226	C₂×M₁₆	C₂³×C₇	1	1	yes	characteristic
227	C₂×M₁₆	C₂³×C₇	1	2	yes	—
228	C₂²×C₈	C₂³×C₇	1	1	yes	characteristic
229	C₂²×C₈	C₂³×C₇	1	1	yes	characteristic
230	C₄×Q₈	—	2	2	no	—
231	C₄×Q₈	—	2	2	no	—
232	C₄×Q₈	—	2	2	no	—
233	C₄×Q₈	—	2	2	no	—
234	C₄×Q₈	C₇×D₄	1	2	yes	—
235	C₄×D₄	—	2	2	no	—
236	C₄×D₄	C₇×D₄	1	2	yes	—
237	#117	—	2	1	no	—
238	#117	C₂³×C₇	1	1	yes	characteristic
239	C₂×C₄⋊C₄	—	2	2	no	—
240	C₂×C₄⋊C₄	C₂³×C₇	1	2	yes	—
241	C₂×C₄²	—	2	1	no	—
242	C₂×C₄²	—	2	2	no	—
243	C₂×C₄²	C₂³×C₇	1	1	yes	characteristic
244	#108	—	2	4	no	—
245	#107	—	2	2	no	—
246	#106	—	2	2	no	—
247	#106	—	2	2	no	—
248	C₄⋊C₈	—	2	2	no	—
249	C₄⋊C₈	—	2	2	no	—
250	#103	—	2	2	no	—
251	#103	—	2	4	no	—
252	#103	—	2	2	no	—
253	#103	—	2	2	no	—
254	#102	—	2	2	no	—
255	#101	—	2	4	no	—
256	#101	—	2	4	no	—
257	C₂²⋊C₈	—	2	2	no	—
258	C₂²⋊C₈	—	2	2	no	—
259	#97	—	2	1	no	—
260	#97	—	2	2	no	—
261	#97	C₇×D₄	1	2	yes	—
262	#97	C₇×D₄	1	2	yes	—
263	C₄×C₈	—	2	1	no	—
264	C₄×C₈	—	2	2	no	—
265	C₂²×C₇	—	2	2	no	—
266	C₂²×C₇	—	4	1	no	—
267	C₂²×C₇	—	4	2	no	—
268	C₂²×C₇	C₂×#121	1	1	yes	characteristic
269	C₂²×C₇	#529	1	2	yes	—
270	C₄×C₇	—	2	2	no	—
271	C₄×C₇	—	2	2	no	—
272	C₄×C₇	—	4	1	no	—
273	C₄×C₇	—	4	4	no	—
274	C₄×C₇	—	4	2	no	—
275	C₄×C₇	—	8	2	no	—
276	C₄×C₇	#525	1	1	yes	characteristic
277	C₄×C₇	#525	1	2	yes	—
278	C₄×C₇	#532	1	1	yes	characteristic
279	#41	—	2	2	no	—
280	#41	—	2	2	no	—
281	C₂×Q₈	—	2	2	no	—
282	C₂×Q₈	—	2	2	no	—
283	C₂×Q₈	—	2	2	no	—
284	C₂×Q₈	—	2	2	no	—
285	C₂×Q₈	—	2	2	no	—
286	C₂×Q₈	C₂×C₇×D₄	1	1	yes	characteristic
287	C₂×Q₈	C₇×#41	1	4	yes	—
288	C₂×D₄	—	2	1	no	—
289	C₂×D₄	C₂×C₇×D₄	1	1	yes	characteristic
290	C₂²×C₄	—	2	2	no	—
291	C₂²×C₄	—	2	1	no	—
292	C₂²×C₄	—	2	2	no	—
293	C₂²×C₄	—	2	2	no	—
294	C₂²×C₄	C₇×#41	1	2	yes	—
295	C₂²×C₄	C₂⁴×C₇	1	1	yes	characteristic
296	Q₁₆	—	2	8	no	—
297	Q₁₆	—	2	8	no	—
298	SD₁₆	—	2	8	no	—
299	M₁₆	—	2	2	no	—
300	M₁₆	—	2	2	no	—
301	M₁₆	—	2	2	no	—
302	M₁₆	—	2	2	no	—
303	M₁₆	—	2	2	no	—
304	M₁₆	C₇×#41	1	4	yes	—
305	C₂×C₈	—	2	2	no	—
306	C₂×C₈	—	2	2	no	—
307	C₂×C₈	—	2	2	no	—
308	C₂×C₈	—	2	1	no	—
309	C₂×C₈	—	2	1	no	—
310	C₂×C₈	—	2	2	no	—
311	C₂×C₈	C₂×C₇×D₄	1	2	yes	—
312	C₂×C₈	C₂×C₇×D₄	1	2	yes	—
313	C₄⋊C₄	—	2	2	no	—
314	C₄⋊C₄	—	2	2	no	—
315	C₄⋊C₄	—	2	2	no	—
316	C₄⋊C₄	—	4	2	no	—
317	C₄⋊C₄	—	4	2	no	—
318	C₄⋊C₄	C₂×C₇×D₄	1	2	yes	—
319	C₂²⋊C₄	—	2	1	no	—
320	C₂²⋊C₄	—	2	2	no	—
321	C₄²	—	2	2	no	—
322	C₄²	—	2	1	no	—
323	C₄²	—	4	2	no	—
324	C₄²	—	4	2	no	—
325	C₄²	C₂×C₇×D₄	1	2	yes	—
326	C₄²	C₂×C₇×D₄	1	1	yes	characteristic
327	C₄²	C₂×C₇×D₄	1	1	yes	characteristic
328	C₂×C₇	—	2	2	no	—
329	C₂×C₇	—	8	1	no	—
330	C₂×C₇	#2392	1	1	yes	characteristic
331	C₂×C₇	#3069	1	2	yes	—
332	C₂³	—	2	1	no	—
333	C₂³	C₂×C₇×#41	1	1	yes	characteristic
334	Q₈	—	2	4	no	—
335	Q₈	—	2	4	no	—
336	Q₈	—	2	2	no	—
337	Q₈	—	4	2	no	—
338	Q₈	—	4	2	no	—
339	D₄	—	2	2	no	—
340	D₄	—	4	2	no	—
341	C₂×C₄	—	2	2	no	—
342	C₂×C₄	—	2	1	no	—
343	C₂×C₄	—	2	4	no	—
344	C₂×C₄	—	4	2	no	—
345	C₂×C₄	—	4	2	no	—
346	C₂×C₄	—	4	2	no	—
347	C₂×C₄	—	4	1	no	—
348	C₂×C₄	—	4	1	no	—
349	C₂×C₄	—	4	2	no	—
350	C₂×C₄	—	8	2	no	—
351	C₂×C₄	—	8	2	no	—
352	C₂×C₄	C₇×#121	1	2	yes	—
353	C₂×C₄	C₇×#121	1	2	yes	—
354	C₂×C₄	C₂²×C₇×D₄	1	1	yes	characteristic
355	C₂×C₄	C₂²×C₇×D₄	1	1	yes	characteristic
356	C₂×C₄	C₂×C₇×#41	1	2	yes	—
357	C₂×C₄	C₂×C₇×#41	1	2	yes	—
358	C₈	—	2	2	no	—
359	C₈	—	2	4	no	—
360	C₈	—	2	2	no	—
361	C₈	—	4	1	no	—
362	C₈	—	4	2	no	—
363	C₈	—	4	1	no	—
364	C₇	#23720	1	1	yes	characteristic, direct product factor, Sylow
365	C₂²	—	2	2	no	—
366	C₂²	—	4	1	no	—
367	C₂²	—	4	2	no	—
368	C₂²	C₂×C₇×#121	1	1	yes	characteristic
369	C₂²	C₇×#529	1	2	yes	—
370	C₄	—	2	2	no	—
371	C₄	—	2	2	no	—
372	C₄	—	4	1	no	—
373	C₄	—	4	4	no	—
374	C₄	—	4	2	no	—
375	C₄	—	8	2	no	—
376	C₄	C₇×#525	1	1	yes	characteristic
377	C₄	C₇×#525	1	2	yes	—
378	C₄	C₇×#532	1	1	yes	characteristic
379	C₂	—	2	2	no	—
380	C₂	—	8	1	no	—
381	C₂	C₇×#2392	1	1	yes	characteristic
382	C₂	C₇×#3069	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₇	C₂×#121
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₇
prime radical	C₂×C₇×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₇ ◁ C₂×C₄²×C₇ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×C₇×#441	C₂	semidirect
4. C₇×#2090	C₂	semidirect
6. C₇×#2064	C₂	semidirect
8. C₇×#2047	C₂	semidirect
10. C₇×#1778	C₂	semidirect
11. C₄×C₇×#101	C₂	semidirect
43. #23720	C₇	direct (N × H)
94. C₂×#441	C₂×C₇	semidirect
96. #2090	C₂×C₇	semidirect
98. #2064	C₂×C₇	semidirect
100. #2047	C₂×C₇	semidirect
102. #1778	C₂×C₇	semidirect
103. C₄×#101	C₂×C₇	semidirect
364. C₇	#23720	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×#137	2
C₂×C₇×#441	2
C₇×#2923	2
C₇×#2090	2
C₇×#2079	2
C₇×#2064	2
C₇×#2052	2
C₇×#2047	2
C₇×#2020	2
C₇×#1778	2
C₄×C₇×#101	2
#23720	7

Subgroups — C7×#23720 #13200936

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×#23720 #13200936