Subgroups — C₇×#22694 #13199910

Order 1792.

812 conjugacy classes of subgroups.

The subgroup lattice has 358 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₇×#2957	C₂	1	2	yes	maximal
2	C₂×C₇×#443	C₂	1	1	yes	maximal, characteristic
3	C₇×#2916	C₂	1	2	yes	maximal, complemented
4	C₇×#1953	C₂	1	2	yes	maximal, complemented
5	C₇×#1861	C₂	1	4	yes	maximal, complemented
6	C₇×#1848	C₂	1	1	yes	maximal, characteristic, complemented
7	C₇×#1814	C₂	1	2	yes	maximal, complemented
8	C₇×#1772	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×C₇×#135	C₂²	1	1	yes	characteristic
10	C₇×#568	C₂²	1	4	yes	—
11	C₂×C₇×#131	C₂²	1	2	yes	—
12	C₇×#444	—	2	4	no	—
13	C₇×#443	—	2	4	no	—
14	C₇×#443	C₄	1	2	yes	complemented
15	C₇×#443	C₄	1	2	yes	complemented
16	C₇×#443	C₄	1	2	yes	complemented
17	C₇×#443	C₄	1	2	yes	complemented
18	C₇×#430	—	2	4	no	—
19	C₇×#430	C₂²	1	4	yes	—
20	C₂×C₇×#108	—	2	4	no	—
21	C₂×C₇×#108	C₂²	1	2	yes	—
22	C₂×C₇×#108	C₂²	1	2	yes	—
23	C₂×C₇×#108	C₂²	1	1	yes	characteristic
24	C₂×C₇×#108	C₂²	1	1	yes	characteristic
25	C₇×#427	C₂²	1	1	yes	characteristic
26	C₇×#423	C₂²	1	2	yes	—
27	C₇×#421	C₂²	1	4	yes	—
28	C₂×C₇×#104	C₂²	1	2	yes	—
29	C₇×#405	C₂²	1	2	yes	—
30	C₂×C₄×C₇×C₈	C₂²	1	1	yes	characteristic
31	C₇×#344	C₂²	1	4	yes	—
32	C₇×#339	C₂²	1	2	yes	—
33	#22694	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
34	C₂×C₇×#41	C₂³	1	2	yes	—
35	C₇×#135	C₂×C₄	1	2	yes	—
36	C₇×#135	C₂×C₄	1	2	yes	—
37	C₇×#135	D₄	1	2	yes	—
38	C₇×#135	D₄	1	2	yes	—
39	C₂×C₇×Q₁₆	Q₈	1	1	yes	characteristic
40	C₂×C₇×SD₁₆	Q₈	1	2	yes	—
41	C₂×C₇×D₈	Q₈	1	1	yes	characteristic
42	C₇×#131	—	2	2	no	—
43	C₇×#131	—	2	2	no	—
44	C₇×#131	—	2	8	no	—
45	C₇×#131	—	4	4	no	—
46	C₇×#131	C₂×C₄	1	4	yes	—
47	C₇×#131	C₂×C₄	1	4	yes	—
48	C₂×C₇×M₁₆	—	2	2	no	—
49	C₂×C₇×M₁₆	—	2	8	no	—
50	C₂×C₇×M₁₆	C₂³	1	4	yes	—
51	C₂×C₇×M₁₆	C₂³	1	2	yes	—
52	C₂×C₇×M₁₆	C₂³	1	2	yes	—
53	C₂²×C₇×C₈	—	2	2	no	—
54	C₂²×C₇×C₈	C₂³	1	1	yes	characteristic
55	C₂²×C₇×C₈	C₂³	1	1	yes	characteristic
56	C₇×#117	C₂³	1	2	yes	—
57	C₂×C₄²×C₇	C₂³	1	1	yes	characteristic
58	C₇×#108	—	2	8	no	—
59	C₇×#108	—	2	2	no	—
60	C₇×#108	—	2	2	no	—
61	C₇×#108	—	2	2	no	—
62	C₇×#108	—	2	2	no	—
63	C₇×#108	C₂×C₄	1	2	yes	—
64	C₇×#108	C₂×C₄	1	2	yes	—
65	C₇×#108	D₄	1	2	yes	—
66	C₇×#108	D₄	1	2	yes	—
67	C₇×#107	—	2	1	no	—
68	C₇×#106	—	2	1	no	—
69	C₇×C₄⋊C₈	—	4	2	no	—
70	C₇×#104	—	2	8	no	—
71	C₇×#104	C₂×C₄	1	4	yes	—
72	C₇×#104	C₂×C₄	1	4	yes	—
73	C₇×#97	—	2	2	no	—
74	C₄×C₇×C₈	—	2	2	no	—
75	C₄×C₇×C₈	C₂×C₄	1	1	yes	characteristic
76	C₄×C₇×C₈	C₂×C₄	1	1	yes	characteristic
77	C₄×C₇×C₈	C₂×C₄	1	1	yes	characteristic
78	C₄×C₇×C₈	C₂×C₄	1	1	yes	characteristic
79	#2957	C₂×C₇	1	2	yes	—
80	C₂×#443	C₂×C₇	1	1	yes	characteristic
81	#2916	C₂×C₇	1	2	yes	complemented
82	#1953	C₂×C₇	1	2	yes	complemented
83	#1861	C₂×C₇	1	4	yes	complemented
84	#1848	C₂×C₇	1	1	yes	characteristic, complemented
85	#1814	C₂×C₇	1	2	yes	complemented
86	#1772	C₂×C₇	1	1	yes	characteristic, complemented
87	C₇×#41	—	2	2	no	—
88	C₇×#41	—	2	2	no	—
89	C₇×#41	C₂²×C₄	1	4	yes	—
90	C₇×#41	C₂×D₄	1	4	yes	—
91	C₂×C₇×Q₈	C₂×Q₈	1	2	yes	—
92	C₂×C₇×D₄	—	2	2	no	—
93	C₂×C₇×D₄	C₂×Q₈	1	2	yes	—
94	C₂²×C₄×C₇	—	2	2	no	—
95	C₂²×C₄×C₇	—	2	1	no	—
96	C₂²×C₄×C₇	C₂⁴	1	1	yes	characteristic
97	C₇×Q₁₆	C₄⋊C₄	1	2	yes	—
98	C₇×Q₁₆	C₄⋊C₄	1	2	yes	—
99	C₇×SD₁₆	C₄⋊C₄	1	4	yes	—
100	C₇×SD₁₆	C₄⋊C₄	1	4	yes	—
101	C₇×D₈	C₄⋊C₄	1	2	yes	—
102	C₇×D₈	C₄⋊C₄	1	2	yes	—
103	C₇×M₁₆	—	2	2	no	—
104	C₇×M₁₆	—	2	2	no	—
105	C₇×M₁₆	—	2	4	no	—
106	C₇×M₁₆	—	2	4	no	—
107	C₇×M₁₆	—	2	4	no	—
108	C₇×M₁₆	—	2	4	no	—
109	C₇×M₁₆	—	4	8	no	—
110	C₇×M₁₆	C₂²×C₄	1	4	yes	—
111	C₇×M₁₆	C₂×D₄	1	4	yes	—
112	C₂×C₇×C₈	—	2	2	no	—
113	C₂×C₇×C₈	—	2	2	no	—
114	C₂×C₇×C₈	—	2	4	no	—
115	C₂×C₇×C₈	—	2	4	no	—
116	C₂×C₇×C₈	—	2	4	no	—
117	C₂×C₇×C₈	—	4	8	no	—
118	C₂×C₇×C₈	C₂²×C₄	1	1	yes	characteristic
119	C₂×C₇×C₈	C₂²×C₄	1	1	yes	characteristic
120	C₂×C₇×C₈	C₂²×C₄	1	1	yes	characteristic
121	C₂×C₇×C₈	C₂²×C₄	1	1	yes	characteristic
122	C₂×C₇×C₈	C₂×D₄	1	1	yes	characteristic
123	C₂×C₇×C₈	C₂×D₄	1	1	yes	characteristic
124	C₂×C₇×C₈	C₂×D₄	1	1	yes	characteristic
125	C₂×C₇×C₈	C₂×D₄	1	1	yes	characteristic
126	C₂×C₇×C₈	C₂×Q₈	1	1	yes	characteristic
127	C₂×C₇×C₈	C₂×Q₈	1	1	yes	characteristic
128	C₂×C₇×C₈	#41	1	1	yes	characteristic
129	C₂×C₇×C₈	#41	1	1	yes	characteristic
130	C₇×C₄⋊C₄	—	2	2	no	—
131	C₇×C₂²⋊C₄	—	2	2	no	—
132	C₄²×C₇	—	2	1	no	—
133	C₄²×C₇	—	2	2	no	—
134	C₄²×C₇	C₂²×C₄	1	1	yes	characteristic
135	C₄²×C₇	C₂²×C₄	1	1	yes	characteristic
136	C₂×#135	C₂²×C₇	1	1	yes	characteristic
137	#568	C₂²×C₇	1	4	yes	—
138	C₂×#131	C₂²×C₇	1	2	yes	—
139	#444	—	2	4	no	—
140	#443	—	2	4	no	—
141	#443	C₄×C₇	1	2	yes	complemented
142	#443	C₄×C₇	1	2	yes	complemented
143	#443	C₄×C₇	1	2	yes	complemented
144	#443	C₄×C₇	1	2	yes	complemented
145	#430	—	2	4	no	—
146	#430	C₂²×C₇	1	4	yes	—
147	C₂×#108	—	2	4	no	—
148	C₂×#108	C₂²×C₇	1	2	yes	—
149	C₂×#108	C₂²×C₇	1	2	yes	—
150	C₂×#108	C₂²×C₇	1	1	yes	characteristic
151	C₂×#108	C₂²×C₇	1	1	yes	characteristic
152	#427	C₂²×C₇	1	1	yes	characteristic
153	#423	C₂²×C₇	1	2	yes	—
154	#421	C₂²×C₇	1	4	yes	—
155	C₂×#104	C₂²×C₇	1	2	yes	—
156	#405	C₂²×C₇	1	2	yes	—
157	C₂×C₄×C₈	C₂²×C₇	1	1	yes	characteristic
158	#344	C₂²×C₇	1	4	yes	—
159	#339	C₂²×C₇	1	2	yes	—
160	C₂³×C₇	—	2	2	no	—
161	C₂³×C₇	2₋¹⁺⁴	1	1	yes	characteristic
162	C₇×Q₈	—	2	2	no	—
163	C₇×Q₈	C₂×C₄⋊C₄	1	4	yes	—
164	C₇×D₄	—	2	4	no	—
165	C₇×D₄	—	2	4	no	—
166	C₇×D₄	—	2	2	no	—
167	C₇×D₄	C₂×C₄⋊C₄	1	4	yes	—
168	C₂×C₄×C₇	—	2	1	no	—
169	C₂×C₄×C₇	—	2	2	no	—
170	C₂×C₄×C₇	—	2	4	no	—
171	C₂×C₄×C₇	—	2	4	no	—
172	C₂×C₄×C₇	—	2	1	no	—
173	C₂×C₄×C₇	—	4	2	no	—
174	C₂×C₄×C₇	—	4	2	no	—
175	C₂×C₄×C₇	C₂³×C₄	1	1	yes	characteristic
176	C₂×C₄×C₇	C₂²×D₄	1	1	yes	characteristic
177	C₂×C₄×C₇	C₂²×D₄	1	1	yes	characteristic
178	C₂×C₄×C₇	C₂²×Q₈	1	1	yes	characteristic
179	C₂×C₄×C₇	C₂×#41	1	1	yes	characteristic
180	C₂×C₄×C₇	H(5,𝔽₂)	1	1	yes	characteristic
181	C₇×C₈	—	2	4	no	—
182	C₇×C₈	—	2	4	no	—
183	C₇×C₈	—	4	4	no	—
184	C₇×C₈	C₂×C₄⋊C₄	1	1	yes	characteristic
185	C₇×C₈	C₂×C₄⋊C₄	1	1	yes	characteristic
186	C₇×C₈	C₂×C₄⋊C₄	1	1	yes	characteristic
187	C₇×C₈	C₂×C₄⋊C₄	1	1	yes	characteristic
188	C₇×C₈	C₄×D₄	1	1	yes	characteristic
189	C₇×C₈	C₄×D₄	1	1	yes	characteristic
190	C₇×C₈	C₄×D₄	1	1	yes	characteristic
191	C₇×C₈	C₄×D₄	1	1	yes	characteristic
192	C₂×#41	C₂³×C₇	1	2	yes	—
193	#135	C₂×C₄×C₇	1	2	yes	—
194	#135	C₂×C₄×C₇	1	2	yes	—
195	#135	C₇×D₄	1	2	yes	—
196	#135	C₇×D₄	1	2	yes	—
197	C₂×Q₁₆	C₇×Q₈	1	1	yes	characteristic
198	C₂×SD₁₆	C₇×Q₈	1	2	yes	—
199	C₂×D₈	C₇×Q₈	1	1	yes	characteristic
200	#131	—	2	2	no	—
201	#131	—	2	2	no	—
202	#131	—	2	8	no	—
203	#131	—	4	4	no	—
204	#131	C₂×C₄×C₇	1	4	yes	—
205	#131	C₂×C₄×C₇	1	4	yes	—
206	C₂×M₁₆	—	2	2	no	—
207	C₂×M₁₆	—	2	8	no	—
208	C₂×M₁₆	C₂³×C₇	1	4	yes	—
209	C₂×M₁₆	C₂³×C₇	1	2	yes	—
210	C₂×M₁₆	C₂³×C₇	1	2	yes	—
211	C₂²×C₈	—	2	2	no	—
212	C₂²×C₈	C₂³×C₇	1	1	yes	characteristic
213	C₂²×C₈	C₂³×C₇	1	1	yes	characteristic
214	#117	C₂³×C₇	1	2	yes	—
215	C₂×C₄²	C₂³×C₇	1	1	yes	characteristic
216	#108	—	2	8	no	—
217	#108	—	2	2	no	—
218	#108	—	2	2	no	—
219	#108	—	2	2	no	—
220	#108	—	2	2	no	—
221	#108	C₂×C₄×C₇	1	2	yes	—
222	#108	C₂×C₄×C₇	1	2	yes	—
223	#108	C₇×D₄	1	2	yes	—
224	#108	C₇×D₄	1	2	yes	—
225	#107	—	2	1	no	—
226	#106	—	2	1	no	—
227	C₄⋊C₈	—	4	2	no	—
228	#104	—	2	8	no	—
229	#104	C₂×C₄×C₇	1	4	yes	—
230	#104	C₂×C₄×C₇	1	4	yes	—
231	#97	—	2	2	no	—
232	C₄×C₈	—	2	2	no	—
233	C₄×C₈	C₂×C₄×C₇	1	1	yes	characteristic
234	C₄×C₈	C₂×C₄×C₇	1	1	yes	characteristic
235	C₄×C₈	C₂×C₄×C₇	1	1	yes	characteristic
236	C₄×C₈	C₂×C₄×C₇	1	1	yes	characteristic
237	C₂²×C₇	—	2	4	no	—
238	C₂²×C₇	—	4	1	no	—
239	C₂²×C₇	—	8	2	no	—
240	C₂²×C₇	#520	1	1	yes	characteristic
241	C₂²×C₇	#547	1	1	yes	characteristic
242	C₂²×C₇	D₄×Q₈	1	1	yes	characteristic
243	C₄×C₇	—	2	2	no	—
244	C₄×C₇	—	2	4	no	—
245	C₄×C₇	—	4	1	no	—
246	C₄×C₇	—	8	2	no	—
247	C₄×C₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
248	C₄×C₇	C₂×C₄×D₄	1	1	yes	characteristic
249	C₄×C₇	D₄²	1	1	yes	characteristic
250	C₄×C₇	#554	1	1	yes	characteristic
251	#41	—	2	2	no	—
252	#41	—	2	2	no	—
253	#41	C₂²×C₄×C₇	1	4	yes	—
254	#41	C₂×C₇×D₄	1	4	yes	—
255	C₂×Q₈	C₂×C₇×Q₈	1	2	yes	—
256	C₂×D₄	—	2	2	no	—
257	C₂×D₄	C₂×C₇×Q₈	1	2	yes	—
258	C₂²×C₄	—	2	2	no	—
259	C₂²×C₄	—	2	1	no	—
260	C₂²×C₄	C₂⁴×C₇	1	1	yes	characteristic
261	Q₁₆	C₇×C₄⋊C₄	1	2	yes	—
262	Q₁₆	C₇×C₄⋊C₄	1	2	yes	—
263	SD₁₆	C₇×C₄⋊C₄	1	4	yes	—
264	SD₁₆	C₇×C₄⋊C₄	1	4	yes	—
265	D₈	C₇×C₄⋊C₄	1	2	yes	—
266	D₈	C₇×C₄⋊C₄	1	2	yes	—
267	M₁₆	—	2	2	no	—
268	M₁₆	—	2	2	no	—
269	M₁₆	—	2	4	no	—
270	M₁₆	—	2	4	no	—
271	M₁₆	—	2	4	no	—
272	M₁₆	—	2	4	no	—
273	M₁₆	—	4	8	no	—
274	M₁₆	C₂²×C₄×C₇	1	4	yes	—
275	M₁₆	C₂×C₇×D₄	1	4	yes	—
276	C₂×C₈	—	2	2	no	—
277	C₂×C₈	—	2	2	no	—
278	C₂×C₈	—	2	4	no	—
279	C₂×C₈	—	2	4	no	—
280	C₂×C₈	—	2	4	no	—
281	C₂×C₈	—	4	8	no	—
282	C₂×C₈	C₂²×C₄×C₇	1	1	yes	characteristic
283	C₂×C₈	C₂²×C₄×C₇	1	1	yes	characteristic
284	C₂×C₈	C₂²×C₄×C₇	1	1	yes	characteristic
285	C₂×C₈	C₂²×C₄×C₇	1	1	yes	characteristic
286	C₂×C₈	C₂×C₇×D₄	1	1	yes	characteristic
287	C₂×C₈	C₂×C₇×D₄	1	1	yes	characteristic
288	C₂×C₈	C₂×C₇×D₄	1	1	yes	characteristic
289	C₂×C₈	C₂×C₇×D₄	1	1	yes	characteristic
290	C₂×C₈	C₂×C₇×Q₈	1	1	yes	characteristic
291	C₂×C₈	C₂×C₇×Q₈	1	1	yes	characteristic
292	C₂×C₈	C₇×#41	1	1	yes	characteristic
293	C₂×C₈	C₇×#41	1	1	yes	characteristic
294	C₄⋊C₄	—	2	2	no	—
295	C₂²⋊C₄	—	2	2	no	—
296	C₄²	—	2	1	no	—
297	C₄²	—	2	2	no	—
298	C₄²	C₂²×C₄×C₇	1	1	yes	characteristic
299	C₄²	C₂²×C₄×C₇	1	1	yes	characteristic
300	C₂×C₇	—	2	1	no	—
301	C₂×C₇	—	2	1	no	—
302	C₂×C₇	—	2	1	no	—
303	C₂×C₇	—	4	4	no	—
304	C₂×C₇	D₄×C₄⋊C₄	1	1	yes	characteristic
305	C₂³	—	2	2	no	—
306	C₂³	C₇×2₋¹⁺⁴	1	1	yes	characteristic
307	Q₈	—	2	2	no	—
308	Q₈	C₂×C₇×C₄⋊C₄	1	4	yes	—
309	D₄	—	2	4	no	—
310	D₄	—	2	4	no	—
311	D₄	—	2	2	no	—
312	D₄	C₂×C₇×C₄⋊C₄	1	4	yes	—
313	C₂×C₄	—	2	1	no	—
314	C₂×C₄	—	2	2	no	—
315	C₂×C₄	—	2	4	no	—
316	C₂×C₄	—	2	4	no	—
317	C₂×C₄	—	2	1	no	—
318	C₂×C₄	—	4	2	no	—
319	C₂×C₄	—	4	2	no	—
320	C₂×C₄	C₂³×C₄×C₇	1	1	yes	characteristic
321	C₂×C₄	C₂²×C₇×D₄	1	1	yes	characteristic
322	C₂×C₄	C₂²×C₇×D₄	1	1	yes	characteristic
323	C₂×C₄	C₂²×C₇×Q₈	1	1	yes	characteristic
324	C₂×C₄	C₂×C₇×#41	1	1	yes	characteristic
325	C₂×C₄	C₇×H(5,𝔽₂)	1	1	yes	characteristic
326	C₈	—	2	4	no	—
327	C₈	—	2	4	no	—
328	C₈	—	4	4	no	—
329	C₈	C₂×C₇×C₄⋊C₄	1	1	yes	characteristic
330	C₈	C₂×C₇×C₄⋊C₄	1	1	yes	characteristic
331	C₈	C₂×C₇×C₄⋊C₄	1	1	yes	characteristic
332	C₈	C₂×C₇×C₄⋊C₄	1	1	yes	characteristic
333	C₈	C₄×C₇×D₄	1	1	yes	characteristic
334	C₈	C₄×C₇×D₄	1	1	yes	characteristic
335	C₈	C₄×C₇×D₄	1	1	yes	characteristic
336	C₈	C₄×C₇×D₄	1	1	yes	characteristic
337	C₇	#22694	1	1	yes	characteristic, direct product factor, Sylow
338	C₂²	—	2	4	no	—
339	C₂²	—	4	1	no	—
340	C₂²	—	8	2	no	—
341	C₂²	C₇×#520	1	1	yes	characteristic
342	C₂²	C₇×#547	1	1	yes	characteristic
343	C₂²	C₇×D₄×Q₈	1	1	yes	characteristic
344	C₄	—	2	2	no	—
345	C₄	—	2	4	no	—
346	C₄	—	4	1	no	—
347	C₄	—	8	2	no	—
348	C₄	C₂²×C₇×C₄⋊C₄	1	1	yes	characteristic
349	C₄	C₂×C₄×C₇×D₄	1	1	yes	characteristic
350	C₄	C₇×D₄²	1	1	yes	characteristic
351	C₄	C₇×#554	1	1	yes	characteristic
352	C₂	—	2	1	no	—
353	C₂	—	2	1	no	—
354	C₂	—	2	1	no	—
355	C₂	—	4	4	no	—
356	C₂	C₇×D₄×C₄⋊C₄	1	1	yes	characteristic

	subgroup	G/H
center	C₄×C₇	D₄²
commutator	C₂×C₄	C₂³×C₄×C₇
Frattini, cosocle	C₂²×C₄	C₂⁴×C₇
socle	C₂×C₇	D₄×C₄⋊C₄
prime radical	C₂×C₇×#135	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄×C₇ ◁ C₂²×C₄×C₇ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₇×#2916	C₂	semidirect
4. C₇×#1953	C₂	semidirect
5. C₇×#1861	C₂	semidirect
6. C₇×#1848	C₂	semidirect
7. C₇×#1814	C₂	semidirect
8. C₇×#1772	C₂	semidirect
14. C₇×#443	C₄	semidirect
15. C₇×#443	C₄	semidirect
16. C₇×#443	C₄	semidirect
17. C₇×#443	C₄	semidirect
33. #22694	C₇	direct (N × H)
81. #2916	C₂×C₇	semidirect
82. #1953	C₂×C₇	semidirect
83. #1861	C₂×C₇	semidirect
84. #1848	C₂×C₇	semidirect
85. #1814	C₂×C₇	semidirect
86. #1772	C₂×C₇	semidirect
141. #443	C₄×C₇	semidirect
142. #443	C₄×C₇	semidirect
143. #443	C₄×C₇	semidirect
144. #443	C₄×C₇	semidirect
337. C₇	#22694	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×#2957	2
C₂×C₇×#443	2
C₇×#2916	2
C₇×#1953	2
C₇×#1861	2
C₇×#1848	2
C₇×#1814	2
C₇×#1772	2
#22694	7

Subgroups — C7×#22694 #13199910

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×#22694 #13199910