Subgroups — C₇×#22692 #13199908

Order 1792.

862 conjugacy classes of subgroups.

The subgroup lattice has 340 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₇×#443	C₂	1	1	yes	maximal, characteristic
2	C₇×#2951	C₂	1	2	yes	maximal
3	C₇×#2913	C₂	1	2	yes	maximal, complemented
4	C₇×#1952	C₂	1	2	yes	maximal, complemented
5	C₇×#1859	C₂	1	4	yes	maximal, complemented
6	C₇×#1845	C₂	1	1	yes	maximal, characteristic, complemented
7	C₇×#1816	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	C₂×C₇×#131	C₂²	1	2	yes	—
11	C₇×#520	C₂²	1	4	yes	—
12	C₇×#443	C₄	1	8	yes	complemented
13	C₇×#440	—	2	4	no	—
14	C₄×C₇×Q₁₆	—	2	2	no	—
15	C₄×C₇×D₈	—	2	2	no	—
16	C₂×C₇×#108	C₂²	1	1	yes	characteristic
17	C₂×C₇×#108	C₂²	1	1	yes	characteristic
18	C₇×#428	—	2	4	no	—
19	C₇×#428	C₂²	1	4	yes	—
20	C₇×#427	—	2	2	no	—
21	C₇×#427	C₂²	1	1	yes	characteristic
22	C₂×C₇×#107	—	2	2	no	—
23	C₂×C₇×#107	C₂²	1	2	yes	—
24	C₂×C₇×#107	C₂²	1	2	yes	—
25	C₂×C₇×#104	C₂²	1	2	yes	—
26	C₇×#417	C₂²	1	4	yes	—
27	C₇×#405	C₂²	1	2	yes	—
28	C₂×C₄×C₇×C₈	C₂²	1	1	yes	characteristic
29	C₇×#384	C₂²	1	2	yes	—
30	C₇×#343	C₂²	1	4	yes	—
31	C₇×#339	C₂²	1	2	yes	—
32	#22692	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
33	C₂×C₇×#41	C₂³	1	2	yes	—
34	C₇×#135	C₂×C₄	1	4	yes	—
35	C₇×#135	Q₈	1	4	yes	—
36	C₂×C₇×Q₁₆	D₄	1	1	yes	characteristic
37	C₂×C₇×SD₁₆	D₄	1	2	yes	—
38	C₂×C₇×D₈	D₄	1	1	yes	characteristic
39	C₇×#131	—	2	4	no	—
40	C₇×#131	C₂×C₄	1	8	yes	complemented
41	C₂×C₇×M₁₆	—	2	2	no	—
42	C₂×C₇×M₁₆	C₂³	1	2	yes	—
43	C₂×C₇×M₁₆	C₂³	1	2	yes	—
44	C₂²×C₇×C₈	—	2	2	no	—
45	C₂²×C₇×C₈	C₂³	1	1	yes	characteristic
46	C₂²×C₇×C₈	C₂³	1	1	yes	characteristic
47	C₄×C₇×Q₈	—	2	4	no	—
48	C₄×C₇×D₄	—	2	4	no	—
49	C₄×C₇×D₄	—	4	4	no	—
50	C₇×#117	—	2	4	no	—
51	C₇×#117	C₂³	1	2	yes	—
52	C₂×C₇×C₄⋊C₄	—	2	4	no	—
53	C₂×C₇×C₄⋊C₄	—	4	2	no	—
54	C₂×C₇×C₄⋊C₄	C₂³	1	4	yes	—
55	C₂×C₄²×C₇	C₂³	1	1	yes	characteristic
56	C₇×#108	—	2	2	no	—
57	C₇×#108	C₂×C₄	1	4	yes	complemented
58	C₇×#107	—	2	2	no	—
59	C₇×#107	—	2	2	no	—
60	C₇×#107	—	2	2	no	—
61	C₇×#107	—	2	2	no	—
62	C₇×#107	—	2	4	no	—
63	C₇×#107	D₄	1	1	yes	characteristic
64	C₇×#107	D₄	1	1	yes	characteristic
65	C₇×#106	—	2	4	no	—
66	C₇×#106	D₄	1	2	yes	—
67	C₇×#104	C₂×C₄	1	8	yes	complemented
68	C₇×#103	—	2	4	no	—
69	C₇×#102	—	2	4	no	—
70	C₇×#97	—	2	2	no	—
71	C₄×C₇×C₈	—	2	2	no	—
72	C₄×C₇×C₈	C₂×C₄	1	4	yes	complemented
73	C₂×#443	C₂×C₇	1	1	yes	characteristic
74	#2951	C₂×C₇	1	2	yes	—
75	#2913	C₂×C₇	1	2	yes	complemented
76	#1952	C₂×C₇	1	2	yes	complemented
77	#1859	C₂×C₇	1	4	yes	complemented
78	#1845	C₂×C₇	1	1	yes	characteristic, complemented
79	#1816	C₂×C₇	1	2	yes	complemented
80	#1772	C₂×C₇	1	1	yes	characteristic, complemented
81	C₇×#41	—	2	2	no	—
82	C₇×#41	—	2	2	no	—
83	C₇×#41	C₂²×C₄	1	4	yes	—
84	C₇×#41	C₂×Q₈	1	4	yes	—
85	C₂×C₇×Q₈	C₂×D₄	1	2	yes	—
86	C₂×C₇×D₄	—	2	2	no	—
87	C₂×C₇×D₄	C₂×D₄	1	2	yes	—
88	C₂²×C₄×C₇	—	2	1	no	—
89	C₂²×C₄×C₇	—	2	2	no	—
90	C₂²×C₄×C₇	—	2	4	no	—
91	C₂²×C₄×C₇	—	4	4	no	—
92	C₂²×C₄×C₇	C₂⁴	1	1	yes	characteristic
93	C₇×Q₁₆	C₄⋊C₄	1	4	yes	complemented
94	C₇×SD₁₆	C₄⋊C₄	1	8	yes	complemented
95	C₇×D₈	C₄⋊C₄	1	4	yes	complemented
96	C₇×M₁₆	—	2	2	no	—
97	C₇×M₁₆	—	2	8	no	—
98	C₇×M₁₆	—	2	2	no	—
99	C₇×M₁₆	—	2	2	no	—
100	C₇×M₁₆	C₂²×C₄	1	4	yes	—
101	C₂×C₇×C₈	—	2	2	no	—
102	C₂×C₇×C₈	—	2	8	no	—
103	C₂×C₇×C₈	—	2	2	no	—
104	C₂×C₇×C₈	C₂²×C₄	1	2	yes	—
105	C₂×C₇×C₈	C₂²×C₄	1	2	yes	—
106	C₂×C₇×C₈	C₂×D₄	1	1	yes	characteristic
107	C₂×C₇×C₈	C₂×D₄	1	1	yes	characteristic
108	C₂×C₇×C₈	C₂×D₄	1	1	yes	characteristic
109	C₂×C₇×C₈	C₂×D₄	1	1	yes	characteristic
110	C₂×C₇×C₈	C₂×Q₈	1	2	yes	—
111	C₂×C₇×C₈	#41	1	2	yes	—
112	C₇×C₄⋊C₄	—	2	4	no	—
113	C₇×C₄⋊C₄	—	2	4	no	—
114	C₇×C₄⋊C₄	—	4	4	no	—
115	C₇×C₄⋊C₄	—	8	4	no	—
116	C₇×C₄⋊C₄	C₂×D₄	1	2	yes	—
117	C₇×C₄⋊C₄	C₂×D₄	1	2	yes	—
118	C₇×C₂²⋊C₄	—	2	2	no	—
119	C₇×C₂²⋊C₄	—	4	4	no	—
120	C₄²×C₇	—	2	2	no	—
121	C₄²×C₇	—	2	1	no	—
122	C₄²×C₇	—	4	4	no	—
123	C₄²×C₇	C₂²×C₄	1	2	yes	—
124	C₂×#135	C₂²×C₇	1	1	yes	characteristic
125	C₂×#131	C₂²×C₇	1	2	yes	—
126	#520	C₂²×C₇	1	4	yes	—
127	#443	C₄×C₇	1	8	yes	complemented
128	#440	—	2	4	no	—
129	C₄×Q₁₆	—	2	2	no	—
130	C₄×D₈	—	2	2	no	—
131	C₂×#108	C₂²×C₇	1	1	yes	characteristic
132	C₂×#108	C₂²×C₇	1	1	yes	characteristic
133	#428	—	2	4	no	—
134	#428	C₂²×C₇	1	4	yes	—
135	#427	—	2	2	no	—
136	#427	C₂²×C₇	1	1	yes	characteristic
137	C₂×#107	—	2	2	no	—
138	C₂×#107	C₂²×C₇	1	2	yes	—
139	C₂×#107	C₂²×C₇	1	2	yes	—
140	C₂×#104	C₂²×C₇	1	2	yes	—
141	#417	C₂²×C₇	1	4	yes	—
142	#405	C₂²×C₇	1	2	yes	—
143	C₂×C₄×C₈	C₂²×C₇	1	1	yes	characteristic
144	#384	C₂²×C₇	1	2	yes	—
145	#343	C₂²×C₇	1	4	yes	—
146	#339	C₂²×C₇	1	2	yes	—
147	C₂³×C₇	—	2	2	no	—
148	C₂³×C₇	H(5,𝔽₂)	1	1	yes	characteristic
149	C₇×Q₈	—	2	2	no	—
150	C₇×Q₈	C₂×C₄⋊C₄	1	4	yes	—
151	C₇×D₄	—	2	4	no	—
152	C₇×D₄	—	2	4	no	—
153	C₇×D₄	—	2	2	no	—
154	C₇×D₄	C₂×C₄⋊C₄	1	4	yes	—
155	C₂×C₄×C₇	—	2	2	no	—
156	C₂×C₄×C₇	—	2	2	no	—
157	C₂×C₄×C₇	—	2	2	no	—
158	C₂×C₄×C₇	—	2	2	no	—
159	C₂×C₄×C₇	—	2	4	no	—
160	C₂×C₄×C₇	—	2	4	no	—
161	C₂×C₄×C₇	—	2	2	no	—
162	C₂×C₄×C₇	—	2	2	no	—
163	C₂×C₄×C₇	—	4	4	no	—
164	C₂×C₄×C₇	—	8	4	no	—
165	C₂×C₄×C₇	—	16	4	no	—
166	C₂×C₄×C₇	C₂³×C₄	1	1	yes	characteristic
167	C₂×C₄×C₇	C₂²×D₄	1	1	yes	characteristic
168	C₂×C₄×C₇	C₂²×D₄	1	1	yes	characteristic
169	C₂×C₄×C₇	C₂²×Q₈	1	1	yes	characteristic
170	C₂×C₄×C₇	C₂×#41	1	1	yes	characteristic
171	C₂×C₄×C₇	2₋¹⁺⁴	1	1	yes	characteristic
172	C₇×C₈	—	2	4	no	—
173	C₇×C₈	—	4	2	no	—
174	C₇×C₈	C₂×C₄⋊C₄	1	2	yes	—
175	C₇×C₈	C₂×C₄⋊C₄	1	2	yes	—
176	C₇×C₈	C₄×D₄	1	2	yes	—
177	C₇×C₈	C₄×D₄	1	2	yes	—
178	C₂×#41	C₂³×C₇	1	2	yes	—
179	#135	C₂×C₄×C₇	1	4	yes	—
180	#135	C₇×Q₈	1	4	yes	—
181	C₂×Q₁₆	C₇×D₄	1	1	yes	characteristic
182	C₂×SD₁₆	C₇×D₄	1	2	yes	—
183	C₂×D₈	C₇×D₄	1	1	yes	characteristic
184	#131	—	2	4	no	—
185	#131	C₂×C₄×C₇	1	8	yes	complemented
186	C₂×M₁₆	—	2	2	no	—
187	C₂×M₁₆	C₂³×C₇	1	2	yes	—
188	C₂×M₁₆	C₂³×C₇	1	2	yes	—
189	C₂²×C₈	—	2	2	no	—
190	C₂²×C₈	C₂³×C₇	1	1	yes	characteristic
191	C₂²×C₈	C₂³×C₇	1	1	yes	characteristic
192	C₄×Q₈	—	2	4	no	—
193	C₄×D₄	—	2	4	no	—
194	C₄×D₄	—	4	4	no	—
195	#117	—	2	4	no	—
196	#117	C₂³×C₇	1	2	yes	—
197	C₂×C₄⋊C₄	—	2	4	no	—
198	C₂×C₄⋊C₄	—	4	2	no	—
199	C₂×C₄⋊C₄	C₂³×C₇	1	4	yes	—
200	C₂×C₄²	C₂³×C₇	1	1	yes	characteristic
201	#108	—	2	2	no	—
202	#108	C₂×C₄×C₇	1	4	yes	complemented
203	#107	—	2	2	no	—
204	#107	—	2	2	no	—
205	#107	—	2	2	no	—
206	#107	—	2	2	no	—
207	#107	—	2	4	no	—
208	#107	C₇×D₄	1	1	yes	characteristic
209	#107	C₇×D₄	1	1	yes	characteristic
210	#106	—	2	4	no	—
211	#106	C₇×D₄	1	2	yes	—
212	#104	C₂×C₄×C₇	1	8	yes	complemented
213	#103	—	2	4	no	—
214	#102	—	2	4	no	—
215	#97	—	2	2	no	—
216	C₄×C₈	—	2	2	no	—
217	C₄×C₈	C₂×C₄×C₇	1	4	yes	complemented
218	C₂²×C₇	—	2	1	no	—
219	C₂²×C₇	—	2	1	no	—
220	C₂²×C₇	—	2	4	no	—
221	C₂²×C₇	—	4	2	no	—
222	C₂²×C₇	—	4	2	no	—
223	C₂²×C₇	#520	1	1	yes	characteristic
224	C₂²×C₇	D₄²	1	1	yes	characteristic
225	C₂²×C₇	#554	1	1	yes	characteristic
226	C₄×C₇	—	2	4	no	—
227	C₄×C₇	—	2	4	no	—
228	C₄×C₇	—	4	2	no	—
229	C₄×C₇	—	4	2	no	—
230	C₄×C₇	—	8	4	no	—
231	C₄×C₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
232	C₄×C₇	C₂×C₄×D₄	1	1	yes	characteristic
233	C₄×C₇	#547	1	1	yes	characteristic
234	C₄×C₇	D₄×Q₈	1	1	yes	characteristic
235	#41	—	2	2	no	—
236	#41	—	2	2	no	—
237	#41	C₂²×C₄×C₇	1	4	yes	—
238	#41	C₂×C₇×Q₈	1	4	yes	—
239	C₂×Q₈	C₂×C₇×D₄	1	2	yes	—
240	C₂×D₄	—	2	2	no	—
241	C₂×D₄	C₂×C₇×D₄	1	2	yes	—
242	C₂²×C₄	—	2	1	no	—
243	C₂²×C₄	—	2	2	no	—
244	C₂²×C₄	—	2	4	no	—
245	C₂²×C₄	—	4	4	no	—
246	C₂²×C₄	C₂⁴×C₇	1	1	yes	characteristic
247	Q₁₆	C₇×C₄⋊C₄	1	4	yes	complemented
248	SD₁₆	C₇×C₄⋊C₄	1	8	yes	complemented
249	D₈	C₇×C₄⋊C₄	1	4	yes	complemented
250	M₁₆	—	2	2	no	—
251	M₁₆	—	2	8	no	—
252	M₁₆	—	2	2	no	—
253	M₁₆	—	2	2	no	—
254	M₁₆	C₂²×C₄×C₇	1	4	yes	—
255	C₂×C₈	—	2	2	no	—
256	C₂×C₈	—	2	8	no	—
257	C₂×C₈	—	2	2	no	—
258	C₂×C₈	C₂²×C₄×C₇	1	2	yes	—
259	C₂×C₈	C₂²×C₄×C₇	1	2	yes	—
260	C₂×C₈	C₂×C₇×D₄	1	1	yes	characteristic
261	C₂×C₈	C₂×C₇×D₄	1	1	yes	characteristic
262	C₂×C₈	C₂×C₇×D₄	1	1	yes	characteristic
263	C₂×C₈	C₂×C₇×D₄	1	1	yes	characteristic
264	C₂×C₈	C₂×C₇×Q₈	1	2	yes	—
265	C₂×C₈	C₇×#41	1	2	yes	—
266	C₄⋊C₄	—	2	4	no	—
267	C₄⋊C₄	—	2	4	no	—
268	C₄⋊C₄	—	4	4	no	—
269	C₄⋊C₄	—	8	4	no	—
270	C₄⋊C₄	C₂×C₇×D₄	1	2	yes	—
271	C₄⋊C₄	C₂×C₇×D₄	1	2	yes	—
272	C₂²⋊C₄	—	2	2	no	—
273	C₂²⋊C₄	—	4	4	no	—
274	C₄²	—	2	2	no	—
275	C₄²	—	2	1	no	—
276	C₄²	—	4	4	no	—
277	C₄²	C₂²×C₄×C₇	1	2	yes	—
278	C₂×C₇	—	2	1	no	—
279	C₂×C₇	—	2	1	no	—
280	C₂×C₇	—	4	4	no	—
281	C₂×C₇	D₄×C₄⋊C₄	1	1	yes	characteristic
282	C₂×C₇	#3292	1	1	yes	characteristic
283	C₂×C₇	#3293	1	1	yes	characteristic
284	C₂³	—	2	2	no	—
285	C₂³	C₇×H(5,𝔽₂)	1	1	yes	characteristic
286	Q₈	—	2	2	no	—
287	Q₈	C₂×C₇×C₄⋊C₄	1	4	yes	—
288	D₄	—	2	4	no	—
289	D₄	—	2	4	no	—
290	D₄	—	2	2	no	—
291	D₄	C₂×C₇×C₄⋊C₄	1	4	yes	—
292	C₂×C₄	—	2	2	no	—
293	C₂×C₄	—	2	2	no	—
294	C₂×C₄	—	2	2	no	—
295	C₂×C₄	—	2	2	no	—
296	C₂×C₄	—	2	4	no	—
297	C₂×C₄	—	2	4	no	—
298	C₂×C₄	—	2	2	no	—
299	C₂×C₄	—	2	2	no	—
300	C₂×C₄	—	4	4	no	—
301	C₂×C₄	—	8	4	no	—
302	C₂×C₄	—	16	4	no	—
303	C₂×C₄	C₂³×C₄×C₇	1	1	yes	characteristic
304	C₂×C₄	C₂²×C₇×D₄	1	1	yes	characteristic
305	C₂×C₄	C₂²×C₇×D₄	1	1	yes	characteristic
306	C₂×C₄	C₂²×C₇×Q₈	1	1	yes	characteristic
307	C₂×C₄	C₂×C₇×#41	1	1	yes	characteristic
308	C₂×C₄	C₇×2₋¹⁺⁴	1	1	yes	characteristic
309	C₈	—	2	4	no	—
310	C₈	—	4	2	no	—
311	C₈	C₂×C₇×C₄⋊C₄	1	2	yes	—
312	C₈	C₂×C₇×C₄⋊C₄	1	2	yes	—
313	C₈	C₄×C₇×D₄	1	2	yes	—
314	C₈	C₄×C₇×D₄	1	2	yes	—
315	C₇	#22692	1	1	yes	characteristic, direct product factor, Sylow
316	C₂²	—	2	1	no	—
317	C₂²	—	2	1	no	—
318	C₂²	—	2	4	no	—
319	C₂²	—	4	2	no	—
320	C₂²	—	4	2	no	—
321	C₂²	C₇×#520	1	1	yes	characteristic
322	C₂²	C₇×D₄²	1	1	yes	characteristic
323	C₂²	C₇×#554	1	1	yes	characteristic
324	C₄	—	2	4	no	—
325	C₄	—	2	4	no	—
326	C₄	—	4	2	no	—
327	C₄	—	4	2	no	—
328	C₄	—	8	4	no	—
329	C₄	C₂²×C₇×C₄⋊C₄	1	1	yes	characteristic
330	C₄	C₂×C₄×C₇×D₄	1	1	yes	characteristic
331	C₄	C₇×#547	1	1	yes	characteristic
332	C₄	C₇×D₄×Q₈	1	1	yes	characteristic
333	C₂	—	2	1	no	—
334	C₂	—	2	1	no	—
335	C₂	—	4	4	no	—
336	C₂	C₇×D₄×C₄⋊C₄	1	1	yes	characteristic
337	C₂	C₇×#3292	1	1	yes	characteristic
338	C₂	C₇×#3293	1	1	yes	characteristic

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

Complements

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

N	H ≅ G/N	type
3. C₇×#2913	C₂	semidirect
4. C₇×#1952	C₂	semidirect
5. C₇×#1859	C₂	semidirect
6. C₇×#1845	C₂	semidirect
7. C₇×#1816	C₂	semidirect
8. C₇×#1772	C₂	semidirect
12. C₇×#443	C₄	semidirect
32. #22692	C₇	direct (N × H)
40. C₇×#131	C₂×C₄	semidirect
57. C₇×#108	C₂×C₄	semidirect
67. C₇×#104	C₂×C₄	semidirect
72. C₄×C₇×C₈	C₂×C₄	semidirect
75. #2913	C₂×C₇	semidirect
76. #1952	C₂×C₇	semidirect
77. #1859	C₂×C₇	semidirect
78. #1845	C₂×C₇	semidirect
79. #1816	C₂×C₇	semidirect
80. #1772	C₂×C₇	semidirect
93. C₇×Q₁₆	C₄⋊C₄	semidirect
94. C₇×SD₁₆	C₄⋊C₄	semidirect
95. C₇×D₈	C₄⋊C₄	semidirect
127. #443	C₄×C₇	semidirect
185. #131	C₂×C₄×C₇	semidirect
202. #108	C₂×C₄×C₇	semidirect
212. #104	C₂×C₄×C₇	semidirect
217. C₄×C₈	C₂×C₄×C₇	semidirect
247. Q₁₆	C₇×C₄⋊C₄	semidirect
248. SD₁₆	C₇×C₄⋊C₄	semidirect
249. D₈	C₇×C₄⋊C₄	semidirect
315. C₇	#22692	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×#443	2
C₇×#2951	2
C₇×#2913	2
C₇×#1952	2
C₇×#1859	2
C₇×#1845	2
C₇×#1816	2
C₇×#1772	2
#22692	7

Subgroups — C7×#22692 #13199908

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×#22692 #13199908