Subgroups — C₇×#33703 #13210919

Order 1792.

570 conjugacy classes of subgroups.

The subgroup lattice has 372 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₇×#3164	C₂	1	1	yes	maximal, characteristic, complemented
2	C₇×#3156	C₂	1	1	yes	maximal, characteristic
3	C₇×#2245	C₂	1	2	yes	maximal, complemented
4	C₇×#2243	C₂	1	2	yes	maximal, complemented
5	C₇×#2242	C₂	1	2	yes	maximal, complemented
6	C₇×#2241	C₂	1	2	yes	maximal, complemented
7	C₇×#2141	C₂	1	1	yes	maximal, characteristic, complemented
8	C₇×#2139	C₂	1	1	yes	maximal, characteristic
9	C₇×#2138	C₂	1	1	yes	maximal, characteristic
10	C₂×C₇×#358	C₂	1	1	yes	maximal, characteristic, complemented
11	C₇×#2108	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×C₇×#137	—	2	1	no	—
13	C₂×C₇×#135	C₂²	1	1	yes	characteristic
14	C₂²×C₇×Q₁₆	C₂²	1	1	yes	characteristic
15	C₇×#544	—	2	1	no	—
16	C₇×#536	—	2	1	no	—
17	C₂×C₇×M₃₂	C₂²	1	1	yes	characteristic
18	C₂²×C₇×C₁₆	C₂²	1	1	yes	characteristic
19	C₇×#501	—	2	2	no	—
20	C₇×#500	C₂²	1	2	yes	—
21	C₇×#499	C₂²	1	2	yes	—
22	C₇×#497	—	2	2	no	—
23	C₇×#495	C₂²	1	2	yes	—
24	C₇×#494	C₂²	1	2	yes	—
25	C₇×#428	—	2	1	no	—
26	C₇×#427	C₂²	1	1	yes	characteristic
27	C₂×C₇×#107	C₂²	1	1	yes	characteristic
28	C₇×#405	C₂²	1	1	yes	characteristic
29	C₇×#358	C₂²	1	1	yes	characteristic
30	C₇×#358	C₂²	1	2	yes	—
31	C₇×#358	C₂²	1	2	yes	—
32	C₇×#358	C₂²	1	2	yes	—
33	C₇×#358	C₂²	1	2	yes	—
34	C₇×#358	C₂²	1	2	yes	—
35	C₇×#358	C₂²	1	1	yes	characteristic
36	C₇×#357	C₂²	1	1	yes	characteristic
37	C₇×#357	C₂²	1	2	yes	—
38	C₇×#357	C₂²	1	1	yes	characteristic
39	C₇×#346	C₂²	1	2	yes	—
40	C₄×C₇×C₁₆	C₂²	1	2	yes	—
41	#33703	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
42	C₂×C₇×#41	—	2	1	no	—
43	C₂²×C₇×Q₈	—	2	1	no	—
44	C₇×#137	—	4	4	no	—
45	C₇×#135	—	2	2	no	—
46	C₇×#135	—	2	2	no	—
47	C₂×C₇×Q₁₆	—	2	2	no	—
48	C₂×C₇×Q₁₆	—	2	1	no	—
49	C₂×C₇×Q₁₆	—	2	2	no	—
50	C₂×C₇×Q₁₆	—	2	2	no	—
51	C₂×C₇×Q₁₆	C₂³	1	1	yes	characteristic
52	C₂×C₇×Q₁₆	C₂³	1	2	yes	—
53	C₂×C₇×SD₁₆	—	2	2	no	—
54	C₂×C₇×SD₁₆	—	2	1	no	—
55	C₂×C₇×D₈	C₂³	1	1	yes	characteristic
56	C₂×C₇×M₁₆	D₄	1	1	yes	characteristic
57	C₂²×C₇×C₈	C₂³	1	1	yes	characteristic
58	C₇×C₄⋊Q₈	—	2	2	no	—
59	C₇×C₄⋊Q₈	—	2	2	no	—
60	C₇×C₄⋊Q₈	—	4	1	no	—
61	C₇×#125	—	2	2	no	—
62	C₇×#125	—	4	1	no	—
63	C₇×#124	—	2	2	no	—
64	C₇×#123	—	4	2	no	—
65	C₇×C₂²⋊Q₈	—	4	1	no	—
66	C₇×C₂²⋊Q₈	—	4	1	no	—
67	C₇×C₂²⋊Q₈	—	4	2	no	—
68	C₇×#117	—	2	1	no	—
69	C₇×#117	D₄	1	1	yes	characteristic
70	C₂×C₇×C₄⋊C₄	—	2	1	no	—
71	C₇×M₃₂	—	2	2	no	—
72	C₂×C₇×C₁₆	—	2	2	no	—
73	C₂×C₇×C₁₆	C₂³	1	1	yes	characteristic
74	C₂×C₇×C₁₆	C₂³	1	1	yes	characteristic
75	C₂×C₇×C₁₆	C₂³	1	2	yes	—
76	C₇×#107	—	2	2	no	—
77	C₇×#107	—	2	1	no	—
78	C₇×#107	C₂³	1	1	yes	characteristic
79	C₇×#107	C₂³	1	2	yes	—
80	C₇×#107	C₂³	1	1	yes	characteristic
81	C₇×#106	—	2	2	no	—
82	C₇×#106	—	2	1	no	—
83	C₇×#97	D₄	1	2	yes	—
84	C₄×C₇×C₈	C₂³	1	2	yes	—
85	#3164	C₂×C₇	1	1	yes	characteristic, complemented
86	#3156	C₂×C₇	1	1	yes	characteristic
87	#2245	C₂×C₇	1	2	yes	complemented
88	#2243	C₂×C₇	1	2	yes	complemented
89	#2242	C₂×C₇	1	2	yes	complemented
90	#2241	C₂×C₇	1	2	yes	complemented
91	#2141	C₂×C₇	1	1	yes	characteristic, complemented
92	#2139	C₂×C₇	1	1	yes	characteristic
93	#2138	C₂×C₇	1	1	yes	characteristic
94	C₂×#358	C₂×C₇	1	1	yes	characteristic, complemented
95	#2108	C₂×C₇	1	1	yes	characteristic, complemented
96	C₇×#41	—	4	2	no	—
97	C₇×#41	—	4	2	no	—
98	C₂×C₇×Q₈	—	2	2	no	—
99	C₂×C₇×Q₈	—	2	1	no	—
100	C₂×C₇×Q₈	—	4	1	no	—
101	C₂×C₇×Q₈	—	4	2	no	—
102	C₂×C₇×Q₈	—	4	2	no	—
103	C₂×C₇×Q₈	—	4	2	no	—
104	C₂×C₇×D₄	—	2	1	no	—
105	C₂×C₇×D₄	—	4	1	no	—
106	C₂²×C₄×C₇	—	4	1	no	—
107	C₂²×C₄×C₇	—	4	1	no	—
108	C₂²×C₄×C₇	—	4	1	no	—
109	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
110	C₇×Q₁₆	—	2	2	no	—
111	C₇×Q₁₆	—	2	2	no	—
112	C₇×Q₁₆	—	2	1	no	—
113	C₇×Q₁₆	—	2	1	no	—
114	C₇×Q₁₆	—	4	2	no	—
115	C₇×Q₁₆	—	4	4	no	—
116	C₇×SD₁₆	—	4	2	no	—
117	C₇×SD₁₆	—	4	4	no	—
118	C₇×D₈	—	2	1	no	—
119	C₇×D₈	—	2	1	no	—
120	C₇×M₁₆	SD₁₆	1	4	yes	—
121	C₂×C₇×C₈	C₂×D₄	1	1	yes	characteristic
122	C₂×C₇×C₈	C₂×D₄	1	2	yes	—
123	C₂×C₇×C₈	#41	1	2	yes	—
124	C₂×C₇×C₈	#41	1	2	yes	—
125	C₂×C₇×C₈	C₂⁴	1	1	yes	characteristic
126	C₇×C₄⋊C₄	—	2	1	no	—
127	C₇×C₄⋊C₄	—	2	2	no	—
128	C₇×C₄⋊C₄	—	2	1	no	—
129	C₇×C₄⋊C₄	—	4	1	no	—
130	C₇×C₄⋊C₄	—	4	2	no	—
131	C₇×C₄⋊C₄	—	4	2	no	—
132	C₇×C₄⋊C₄	—	4	2	no	—
133	C₇×C₄⋊C₄	—	4	2	no	—
134	C₇×C₄⋊C₄	—	4	2	no	—
135	C₇×C₄⋊C₄	D₈	1	1	yes	characteristic
136	C₇×C₄⋊C₄	D₈	1	1	yes	characteristic
137	C₇×C₂²⋊C₄	—	4	1	no	—
138	C₇×C₂²⋊C₄	—	4	2	no	—
139	C₇×C₂²⋊C₄	—	4	2	no	—
140	C₇×C₂²⋊C₄	D₈	1	2	yes	—
141	C₄²×C₇	—	4	1	no	—
142	C₄²×C₇	C₂×D₄	1	2	yes	—
143	C₇×C₁₆	—	2	1	no	—
144	C₇×C₁₆	—	2	1	no	—
145	C₇×C₁₆	—	2	2	no	—
146	C₂×#137	—	2	1	no	—
147	C₂×#135	C₂²×C₇	1	1	yes	characteristic
148	C₂²×Q₁₆	C₂²×C₇	1	1	yes	characteristic
149	#544	—	2	1	no	—
150	#536	—	2	1	no	—
151	C₂×M₃₂	C₂²×C₇	1	1	yes	characteristic
152	C₂²×C₁₆	C₂²×C₇	1	1	yes	characteristic
153	#501	—	2	2	no	—
154	#500	C₂²×C₇	1	2	yes	—
155	#499	C₂²×C₇	1	2	yes	—
156	#497	—	2	2	no	—
157	#495	C₂²×C₇	1	2	yes	—
158	#494	C₂²×C₇	1	2	yes	—
159	#428	—	2	1	no	—
160	#427	C₂²×C₇	1	1	yes	characteristic
161	C₂×#107	C₂²×C₇	1	1	yes	characteristic
162	#405	C₂²×C₇	1	1	yes	characteristic
163	#358	C₂²×C₇	1	2	yes	—
164	#358	C₂²×C₇	1	2	yes	—
165	#358	C₂²×C₇	1	1	yes	characteristic
166	#358	C₂²×C₇	1	1	yes	characteristic
167	#358	C₂²×C₇	1	2	yes	—
168	#358	C₂²×C₇	1	2	yes	—
169	#358	C₂²×C₇	1	2	yes	—
170	#357	C₂²×C₇	1	1	yes	characteristic
171	#357	C₂²×C₇	1	1	yes	characteristic
172	#357	C₂²×C₇	1	2	yes	—
173	#346	C₂²×C₇	1	2	yes	—
174	C₄×C₁₆	C₂²×C₇	1	2	yes	—
175	C₂³×C₇	—	4	1	no	—
176	C₂³×C₇	C₂×D₈	1	1	yes	characteristic
177	C₇×Q₈	—	4	1	no	—
178	C₇×Q₈	—	4	2	no	—
179	C₇×Q₈	—	4	2	no	—
180	C₇×Q₈	—	4	1	no	—
181	C₇×Q₈	—	8	2	no	—
182	C₇×Q₈	—	16	2	no	—
183	C₇×D₄	—	4	1	no	—
184	C₇×D₄	—	4	1	no	—
185	C₇×D₄	—	8	2	no	—
186	C₂×C₄×C₇	—	4	1	no	—
187	C₂×C₄×C₇	—	4	1	no	—
188	C₂×C₄×C₇	—	4	2	no	—
189	C₂×C₄×C₇	—	4	2	no	—
190	C₂×C₄×C₇	—	4	1	no	—
191	C₂×C₄×C₇	—	8	2	no	—
192	C₂×C₄×C₇	—	8	2	no	—
193	C₂×C₄×C₇	—	16	1	no	—
194	C₂×C₄×C₇	#124	1	2	yes	—
195	C₂×C₄×C₇	C₂×D₈	1	2	yes	—
196	C₂×C₄×C₇	C₂×D₈	1	2	yes	—
197	C₂×C₄×C₇	C₂×D₈	1	1	yes	characteristic
198	C₂×C₄×C₇	C₂×SD₁₆	1	2	yes	—
199	C₂×C₄×C₇	C₂²×D₄	1	1	yes	characteristic
200	C₇×C₈	#124	1	2	yes	—
201	C₇×C₈	C₂×SD₁₆	1	4	yes	—
202	C₇×C₈	C₂×#41	1	1	yes	characteristic
203	C₇×C₈	C₂×#41	1	1	yes	characteristic
204	C₂×#41	—	2	1	no	—
205	C₂²×Q₈	—	2	1	no	—
206	#137	—	4	4	no	—
207	#135	—	2	2	no	—
208	#135	—	2	2	no	—
209	C₂×Q₁₆	—	2	1	no	—
210	C₂×Q₁₆	—	2	2	no	—
211	C₂×Q₁₆	—	2	2	no	—
212	C₂×Q₁₆	—	2	2	no	—
213	C₂×Q₁₆	C₂³×C₇	1	2	yes	—
214	C₂×Q₁₆	C₂³×C₇	1	1	yes	characteristic
215	C₂×SD₁₆	—	2	1	no	—
216	C₂×SD₁₆	—	2	2	no	—
217	C₂×D₈	C₂³×C₇	1	1	yes	characteristic
218	C₂×M₁₆	C₇×D₄	1	1	yes	characteristic
219	C₂²×C₈	C₂³×C₇	1	1	yes	characteristic
220	C₄⋊Q₈	—	2	2	no	—
221	C₄⋊Q₈	—	2	2	no	—
222	C₄⋊Q₈	—	4	1	no	—
223	#125	—	2	2	no	—
224	#125	—	4	1	no	—
225	#124	—	2	2	no	—
226	#123	—	4	2	no	—
227	C₂²⋊Q₈	—	4	1	no	—
228	C₂²⋊Q₈	—	4	1	no	—
229	C₂²⋊Q₈	—	4	2	no	—
230	#117	—	2	1	no	—
231	#117	C₇×D₄	1	1	yes	characteristic
232	C₂×C₄⋊C₄	—	2	1	no	—
233	M₃₂	—	2	2	no	—
234	C₂×C₁₆	—	2	2	no	—
235	C₂×C₁₆	C₂³×C₇	1	1	yes	characteristic
236	C₂×C₁₆	C₂³×C₇	1	2	yes	—
237	C₂×C₁₆	C₂³×C₇	1	1	yes	characteristic
238	#107	—	2	1	no	—
239	#107	—	2	2	no	—
240	#107	C₂³×C₇	1	2	yes	—
241	#107	C₂³×C₇	1	1	yes	characteristic
242	#107	C₂³×C₇	1	1	yes	characteristic
243	#106	—	2	1	no	—
244	#106	—	2	2	no	—
245	#97	C₇×D₄	1	2	yes	—
246	C₄×C₈	C₂³×C₇	1	2	yes	—
247	C₂²×C₇	—	2	1	no	—
248	C₂²×C₇	—	2	1	no	—
249	C₂²×C₇	—	8	1	no	—
250	C₂²×C₇	—	8	1	no	—
251	C₂²×C₇	—	8	1	no	—
252	C₂²×C₇	#486	1	2	yes	—
253	C₂²×C₇	C₂²×D₈	1	1	yes	characteristic
254	C₄×C₇	—	2	2	no	—
255	C₄×C₇	—	2	2	no	—
256	C₄×C₇	—	8	1	no	—
257	C₄×C₇	—	8	2	no	—
258	C₄×C₇	—	8	2	no	—
259	C₄×C₇	—	8	1	no	—
260	C₄×C₇	—	8	1	no	—
261	C₄×C₇	#486	1	2	yes	—
262	C₄×C₇	C₂×#124	1	1	yes	characteristic
263	C₄×C₇	C₂²×SD₁₆	1	1	yes	characteristic
264	#41	—	4	2	no	—
265	#41	—	4	2	no	—
266	C₂×Q₈	—	2	2	no	—
267	C₂×Q₈	—	2	1	no	—
268	C₂×Q₈	—	4	1	no	—
269	C₂×Q₈	—	4	2	no	—
270	C₂×Q₈	—	4	2	no	—
271	C₂×Q₈	—	4	2	no	—
272	C₂×D₄	—	2	1	no	—
273	C₂×D₄	—	4	1	no	—
274	C₂²×C₄	—	4	1	no	—
275	C₂²×C₄	—	4	1	no	—
276	C₂²×C₄	—	4	1	no	—
277	C₂²×C₄	C₂×C₇×D₄	1	1	yes	characteristic
278	Q₁₆	—	2	2	no	—
279	Q₁₆	—	2	2	no	—
280	Q₁₆	—	2	1	no	—
281	Q₁₆	—	2	1	no	—
282	Q₁₆	—	4	2	no	—
283	Q₁₆	—	4	4	no	—
284	SD₁₆	—	4	2	no	—
285	SD₁₆	—	4	4	no	—
286	D₈	—	2	1	no	—
287	D₈	—	2	1	no	—
288	M₁₆	C₇×SD₁₆	1	4	yes	—
289	C₂×C₈	C₂×C₇×D₄	1	1	yes	characteristic
290	C₂×C₈	C₂×C₇×D₄	1	2	yes	—
291	C₂×C₈	C₇×#41	1	2	yes	—
292	C₂×C₈	C₇×#41	1	2	yes	—
293	C₂×C₈	C₂⁴×C₇	1	1	yes	characteristic
294	C₄⋊C₄	—	2	1	no	—
295	C₄⋊C₄	—	2	2	no	—
296	C₄⋊C₄	—	2	1	no	—
297	C₄⋊C₄	—	4	1	no	—
298	C₄⋊C₄	—	4	2	no	—
299	C₄⋊C₄	—	4	2	no	—
300	C₄⋊C₄	—	4	2	no	—
301	C₄⋊C₄	—	4	2	no	—
302	C₄⋊C₄	—	4	2	no	—
303	C₄⋊C₄	C₇×D₈	1	1	yes	characteristic
304	C₄⋊C₄	C₇×D₈	1	1	yes	characteristic
305	C₂²⋊C₄	—	4	1	no	—
306	C₂²⋊C₄	—	4	2	no	—
307	C₂²⋊C₄	—	4	2	no	—
308	C₂²⋊C₄	C₇×D₈	1	2	yes	—
309	C₄²	—	4	1	no	—
310	C₄²	C₂×C₇×D₄	1	2	yes	—
311	C₁₆	—	2	1	no	—
312	C₁₆	—	2	1	no	—
313	C₁₆	—	2	2	no	—
314	C₂×C₇	—	2	2	no	—
315	C₂×C₇	—	16	1	no	—
316	C₂×C₇	C₂×#486	1	1	yes	characteristic
317	C₂×C₇	#3416	1	1	yes	characteristic
318	C₂×C₇	#3417	1	1	yes	characteristic
319	C₂³	—	4	1	no	—
320	C₂³	C₂×C₇×D₈	1	1	yes	characteristic
321	Q₈	—	4	1	no	—
322	Q₈	—	4	2	no	—
323	Q₈	—	4	2	no	—
324	Q₈	—	4	1	no	—
325	Q₈	—	8	2	no	—
326	Q₈	—	16	2	no	—
327	D₄	—	4	1	no	—
328	D₄	—	4	1	no	—
329	D₄	—	8	2	no	—
330	C₂×C₄	—	4	1	no	—
331	C₂×C₄	—	4	1	no	—
332	C₂×C₄	—	4	2	no	—
333	C₂×C₄	—	4	2	no	—
334	C₂×C₄	—	4	1	no	—
335	C₂×C₄	—	8	2	no	—
336	C₂×C₄	—	8	2	no	—
337	C₂×C₄	—	16	1	no	—
338	C₂×C₄	C₇×#124	1	2	yes	—
339	C₂×C₄	C₂×C₇×D₈	1	2	yes	—
340	C₂×C₄	C₂×C₇×D₈	1	2	yes	—
341	C₂×C₄	C₂×C₇×D₈	1	1	yes	characteristic
342	C₂×C₄	C₂×C₇×SD₁₆	1	2	yes	—
343	C₂×C₄	C₂²×C₇×D₄	1	1	yes	characteristic
344	C₈	C₇×#124	1	2	yes	—
345	C₈	C₂×C₇×SD₁₆	1	4	yes	—
346	C₈	C₂×C₇×#41	1	1	yes	characteristic
347	C₈	C₂×C₇×#41	1	1	yes	characteristic
348	C₇	#33703	1	1	yes	characteristic, direct product factor, Sylow
349	C₂²	—	2	1	no	—
350	C₂²	—	2	1	no	—
351	C₂²	—	8	1	no	—
352	C₂²	—	8	1	no	—
353	C₂²	—	8	1	no	—
354	C₂²	C₇×#486	1	2	yes	—
355	C₂²	C₂²×C₇×D₈	1	1	yes	characteristic
356	C₄	—	2	2	no	—
357	C₄	—	2	2	no	—
358	C₄	—	8	2	no	—
359	C₄	—	8	1	no	—
360	C₄	—	8	1	no	—
361	C₄	—	8	1	no	—
362	C₄	—	8	2	no	—
363	C₄	C₇×#486	1	2	yes	—
364	C₄	C₂×C₇×#124	1	1	yes	characteristic
365	C₄	C₂²×C₇×SD₁₆	1	1	yes	characteristic
366	C₂	—	2	2	no	—
367	C₂	—	16	1	no	—
368	C₂	C₂×C₇×#486	1	1	yes	characteristic
369	C₂	C₇×#3416	1	1	yes	characteristic
370	C₂	C₇×#3417	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₇	C₂²×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂⁴×C₇
prime radical	C₂×C₇×#135	C₂²
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₇ ◁ C₇×#117 ◁ C₇×#405 ◁ G

Complements

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

N	H ≅ G/N	type
1. C₇×#3164	C₂	semidirect
3. C₇×#2245	C₂	semidirect
4. C₇×#2243	C₂	semidirect
5. C₇×#2242	C₂	semidirect
6. C₇×#2241	C₂	semidirect
7. C₇×#2141	C₂	semidirect
10. C₂×C₇×#358	C₂	semidirect
11. C₇×#2108	C₂	semidirect
41. #33703	C₇	direct (N × H)
85. #3164	C₂×C₇	semidirect
87. #2245	C₂×C₇	semidirect
88. #2243	C₂×C₇	semidirect
89. #2242	C₂×C₇	semidirect
90. #2241	C₂×C₇	semidirect
91. #2141	C₂×C₇	semidirect
94. C₂×#358	C₂×C₇	semidirect
95. #2108	C₂×C₇	semidirect
348. C₇	#33703	direct (N × H)

Maximal subgroups

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

subgroup	index
C₇×#3164	2
C₇×#3156	2
C₇×#2245	2
C₇×#2243	2
C₇×#2242	2
C₇×#2241	2
C₇×#2141	2
C₇×#2139	2
C₇×#2138	2
C₂×C₇×#358	2
C₇×#2108	2
#33703	7

Subgroups — C7×#33703 #13210919

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×#33703 #13210919