Subgroups — C₇×#22690 #13199906

Order 1792.

846 conjugacy classes of subgroups.

The subgroup lattice has 330 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×#2912	C₂	1	2	yes	maximal, complemented
4	C₇×#1952	C₂	1	2	yes	maximal, complemented
5	C₇×#1860	C₂	1	4	yes	maximal, complemented
6	C₇×#1846	C₂	1	1	yes	maximal, characteristic, complemented
7	C₇×#1815	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	C₂²	1	2	yes	—
23	C₂×C₇×#106	—	2	2	no	—
24	C₂×C₇×#106	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×#382	C₂²	1	2	yes	—
30	C₇×#343	C₂²	1	4	yes	—
31	C₇×#339	C₂²	1	2	yes	—
32	#22690	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	C₂³	1	4	yes	—
54	C₂×C₄²×C₇	C₂³	1	1	yes	characteristic
55	C₇×#108	—	2	2	no	—
56	C₇×#108	C₂×C₄	1	4	yes	complemented
57	C₇×#107	—	2	2	no	—
58	C₇×#107	—	2	2	no	—
59	C₇×#107	—	2	4	no	—
60	C₇×#107	D₄	1	2	yes	—
61	C₇×#106	—	2	2	no	—
62	C₇×#106	—	2	2	no	—
63	C₇×#106	—	2	4	no	—
64	C₇×#106	D₄	1	1	yes	characteristic
65	C₇×#106	D₄	1	1	yes	characteristic
66	C₇×#104	C₂×C₄	1	8	yes	complemented
67	C₇×#103	—	2	4	no	—
68	C₇×#102	—	2	4	no	—
69	C₇×#97	—	2	2	no	—
70	C₄×C₇×C₈	—	2	2	no	—
71	C₄×C₇×C₈	C₂×C₄	1	4	yes	complemented
72	C₇×#95	—	4	2	no	—
73	C₂×#443	C₂×C₇	1	1	yes	characteristic
74	#2951	C₂×C₇	1	2	yes	—
75	#2912	C₂×C₇	1	2	yes	complemented
76	#1952	C₂×C₇	1	2	yes	complemented
77	#1860	C₂×C₇	1	4	yes	complemented
78	#1846	C₂×C₇	1	1	yes	characteristic, complemented
79	#1815	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	2	no	—
89	C₂²×C₄×C₇	—	2	1	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	—
94	C₇×SD₁₆	C₄⋊C₄	1	8	yes	—
95	C₇×D₈	C₄⋊C₄	1	4	yes	—
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	C₂×D₄	1	2	yes	—
116	C₇×C₄⋊C₄	C₂×D₄	1	2	yes	—
117	C₇×C₂²⋊C₄	—	2	2	no	—
118	C₇×C₂²⋊C₄	—	4	4	no	—
119	C₄²×C₇	—	2	2	no	—
120	C₄²×C₇	—	2	1	no	—
121	C₄²×C₇	—	4	4	no	—
122	C₄²×C₇	C₂²×C₄	1	2	yes	—
123	C₂×#135	C₂²×C₇	1	1	yes	characteristic
124	C₂×#131	C₂²×C₇	1	2	yes	—
125	#520	C₂²×C₇	1	4	yes	—
126	#443	C₄×C₇	1	8	yes	complemented
127	#440	—	2	4	no	—
128	C₄×Q₁₆	—	2	2	no	—
129	C₄×D₈	—	2	2	no	—
130	C₂×#108	C₂²×C₇	1	1	yes	characteristic
131	C₂×#108	C₂²×C₇	1	1	yes	characteristic
132	#428	—	2	4	no	—
133	#428	C₂²×C₇	1	4	yes	—
134	#427	—	2	2	no	—
135	#427	C₂²×C₇	1	1	yes	characteristic
136	C₂×#107	C₂²×C₇	1	2	yes	—
137	C₂×#106	—	2	2	no	—
138	C₂×#106	C₂²×C₇	1	2	yes	—
139	C₂×#104	C₂²×C₇	1	2	yes	—
140	#417	C₂²×C₇	1	4	yes	—
141	#405	C₂²×C₇	1	2	yes	—
142	C₂×C₄×C₈	C₂²×C₇	1	1	yes	characteristic
143	#382	C₂²×C₇	1	2	yes	—
144	#343	C₂²×C₇	1	4	yes	—
145	#339	C₂²×C₇	1	2	yes	—
146	C₂³×C₇	—	2	2	no	—
147	C₂³×C₇	H(5,𝔽₂)	1	1	yes	characteristic
148	C₇×Q₈	—	2	2	no	—
149	C₇×Q₈	C₂×C₄⋊C₄	1	4	yes	—
150	C₇×D₄	—	2	4	no	—
151	C₇×D₄	—	2	4	no	—
152	C₇×D₄	—	2	2	no	—
153	C₇×D₄	C₂×C₄⋊C₄	1	4	yes	—
154	C₂×C₄×C₇	—	2	2	no	—
155	C₂×C₄×C₇	—	2	4	no	—
156	C₂×C₄×C₇	—	2	4	no	—
157	C₂×C₄×C₇	—	2	2	no	—
158	C₂×C₄×C₇	—	2	2	no	—
159	C₂×C₄×C₇	—	2	2	no	—
160	C₂×C₄×C₇	—	4	2	no	—
161	C₂×C₄×C₇	—	4	4	no	—
162	C₂×C₄×C₇	—	8	4	no	—
163	C₂×C₄×C₇	—	16	4	no	—
164	C₂×C₄×C₇	C₂³×C₄	1	1	yes	characteristic
165	C₂×C₄×C₇	C₂²×D₄	1	1	yes	characteristic
166	C₂×C₄×C₇	C₂²×D₄	1	1	yes	characteristic
167	C₂×C₄×C₇	C₂²×Q₈	1	1	yes	characteristic
168	C₂×C₄×C₇	C₂×#41	1	1	yes	characteristic
169	C₂×C₄×C₇	2₋¹⁺⁴	1	1	yes	characteristic
170	C₇×C₈	—	2	4	no	—
171	C₇×C₈	—	4	2	no	—
172	C₇×C₈	C₂×C₄⋊C₄	1	2	yes	—
173	C₇×C₈	C₂×C₄⋊C₄	1	2	yes	—
174	C₇×C₈	C₄×D₄	1	2	yes	—
175	C₇×C₈	C₄×D₄	1	2	yes	—
176	C₂×#41	C₂³×C₇	1	2	yes	—
177	#135	C₂×C₄×C₇	1	4	yes	—
178	#135	C₇×Q₈	1	4	yes	—
179	C₂×Q₁₆	C₇×D₄	1	1	yes	characteristic
180	C₂×SD₁₆	C₇×D₄	1	2	yes	—
181	C₂×D₈	C₇×D₄	1	1	yes	characteristic
182	#131	—	2	4	no	—
183	#131	C₂×C₄×C₇	1	8	yes	complemented
184	C₂×M₁₆	—	2	2	no	—
185	C₂×M₁₆	C₂³×C₇	1	2	yes	—
186	C₂×M₁₆	C₂³×C₇	1	2	yes	—
187	C₂²×C₈	—	2	2	no	—
188	C₂²×C₈	C₂³×C₇	1	1	yes	characteristic
189	C₂²×C₈	C₂³×C₇	1	1	yes	characteristic
190	C₄×Q₈	—	2	4	no	—
191	C₄×D₄	—	2	4	no	—
192	C₄×D₄	—	4	4	no	—
193	#117	—	2	4	no	—
194	#117	C₂³×C₇	1	2	yes	—
195	C₂×C₄⋊C₄	—	2	4	no	—
196	C₂×C₄⋊C₄	C₂³×C₇	1	4	yes	—
197	C₂×C₄²	C₂³×C₇	1	1	yes	characteristic
198	#108	—	2	2	no	—
199	#108	C₂×C₄×C₇	1	4	yes	complemented
200	#107	—	2	2	no	—
201	#107	—	2	2	no	—
202	#107	—	2	4	no	—
203	#107	C₇×D₄	1	2	yes	—
204	#106	—	2	2	no	—
205	#106	—	2	2	no	—
206	#106	—	2	4	no	—
207	#106	C₇×D₄	1	1	yes	characteristic
208	#106	C₇×D₄	1	1	yes	characteristic
209	#104	C₂×C₄×C₇	1	8	yes	complemented
210	#103	—	2	4	no	—
211	#102	—	2	4	no	—
212	#97	—	2	2	no	—
213	C₄×C₈	—	2	2	no	—
214	C₄×C₈	C₂×C₄×C₇	1	4	yes	complemented
215	#95	—	4	2	no	—
216	C₂²×C₇	—	2	2	no	—
217	C₂²×C₇	—	2	4	no	—
218	C₂²×C₇	—	4	4	no	—
219	C₂²×C₇	#520	1	1	yes	characteristic
220	C₂²×C₇	D₄²	1	1	yes	characteristic
221	C₂²×C₇	#554	1	1	yes	characteristic
222	C₄×C₇	—	2	4	no	—
223	C₄×C₇	—	4	2	no	—
224	C₄×C₇	—	4	2	no	—
225	C₄×C₇	—	4	2	no	—
226	C₄×C₇	—	8	4	no	—
227	C₄×C₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
228	C₄×C₇	C₂×C₄×D₄	1	1	yes	characteristic
229	C₄×C₇	#547	1	1	yes	characteristic
230	C₄×C₇	D₄×Q₈	1	1	yes	characteristic
231	#41	—	2	2	no	—
232	#41	—	2	2	no	—
233	#41	C₂²×C₄×C₇	1	4	yes	—
234	#41	C₂×C₇×Q₈	1	4	yes	—
235	C₂×Q₈	C₂×C₇×D₄	1	2	yes	—
236	C₂×D₄	—	2	2	no	—
237	C₂×D₄	C₂×C₇×D₄	1	2	yes	—
238	C₂²×C₄	—	2	2	no	—
239	C₂²×C₄	—	2	1	no	—
240	C₂²×C₄	—	2	4	no	—
241	C₂²×C₄	—	4	4	no	—
242	C₂²×C₄	C₂⁴×C₇	1	1	yes	characteristic
243	Q₁₆	C₇×C₄⋊C₄	1	4	yes	—
244	SD₁₆	C₇×C₄⋊C₄	1	8	yes	—
245	D₈	C₇×C₄⋊C₄	1	4	yes	—
246	M₁₆	—	2	2	no	—
247	M₁₆	—	2	8	no	—
248	M₁₆	—	2	2	no	—
249	M₁₆	—	2	2	no	—
250	M₁₆	C₂²×C₄×C₇	1	4	yes	—
251	C₂×C₈	—	2	2	no	—
252	C₂×C₈	—	2	8	no	—
253	C₂×C₈	—	2	2	no	—
254	C₂×C₈	C₂²×C₄×C₇	1	2	yes	—
255	C₂×C₈	C₂²×C₄×C₇	1	2	yes	—
256	C₂×C₈	C₂×C₇×D₄	1	1	yes	characteristic
257	C₂×C₈	C₂×C₇×D₄	1	1	yes	characteristic
258	C₂×C₈	C₂×C₇×D₄	1	1	yes	characteristic
259	C₂×C₈	C₂×C₇×D₄	1	1	yes	characteristic
260	C₂×C₈	C₂×C₇×Q₈	1	2	yes	—
261	C₂×C₈	C₇×#41	1	2	yes	—
262	C₄⋊C₄	—	2	4	no	—
263	C₄⋊C₄	—	2	4	no	—
264	C₄⋊C₄	—	4	4	no	—
265	C₄⋊C₄	C₂×C₇×D₄	1	2	yes	—
266	C₄⋊C₄	C₂×C₇×D₄	1	2	yes	—
267	C₂²⋊C₄	—	2	2	no	—
268	C₂²⋊C₄	—	4	4	no	—
269	C₄²	—	2	2	no	—
270	C₄²	—	2	1	no	—
271	C₄²	—	4	4	no	—
272	C₄²	C₂²×C₄×C₇	1	2	yes	—
273	C₂×C₇	—	2	1	no	—
274	C₂×C₇	—	2	1	no	—
275	C₂×C₇	—	4	4	no	—
276	C₂×C₇	D₄×C₄⋊C₄	1	1	yes	characteristic
277	C₂×C₇	#3290	1	2	yes	—
278	C₂³	—	2	2	no	—
279	C₂³	C₇×H(5,𝔽₂)	1	1	yes	characteristic
280	Q₈	—	2	2	no	—
281	Q₈	C₂×C₇×C₄⋊C₄	1	4	yes	—
282	D₄	—	2	4	no	—
283	D₄	—	2	4	no	—
284	D₄	—	2	2	no	—
285	D₄	C₂×C₇×C₄⋊C₄	1	4	yes	—
286	C₂×C₄	—	2	2	no	—
287	C₂×C₄	—	2	4	no	—
288	C₂×C₄	—	2	4	no	—
289	C₂×C₄	—	2	2	no	—
290	C₂×C₄	—	2	2	no	—
291	C₂×C₄	—	2	2	no	—
292	C₂×C₄	—	4	2	no	—
293	C₂×C₄	—	4	4	no	—
294	C₂×C₄	—	8	4	no	—
295	C₂×C₄	—	16	4	no	—
296	C₂×C₄	C₂³×C₄×C₇	1	1	yes	characteristic
297	C₂×C₄	C₂²×C₇×D₄	1	1	yes	characteristic
298	C₂×C₄	C₂²×C₇×D₄	1	1	yes	characteristic
299	C₂×C₄	C₂²×C₇×Q₈	1	1	yes	characteristic
300	C₂×C₄	C₂×C₇×#41	1	1	yes	characteristic
301	C₂×C₄	C₇×2₋¹⁺⁴	1	1	yes	characteristic
302	C₈	—	2	4	no	—
303	C₈	—	4	2	no	—
304	C₈	C₂×C₇×C₄⋊C₄	1	2	yes	—
305	C₈	C₂×C₇×C₄⋊C₄	1	2	yes	—
306	C₈	C₄×C₇×D₄	1	2	yes	—
307	C₈	C₄×C₇×D₄	1	2	yes	—
308	C₇	#22690	1	1	yes	characteristic, direct product factor, Sylow
309	C₂²	—	2	2	no	—
310	C₂²	—	2	4	no	—
311	C₂²	—	4	4	no	—
312	C₂²	C₇×#520	1	1	yes	characteristic
313	C₂²	C₇×D₄²	1	1	yes	characteristic
314	C₂²	C₇×#554	1	1	yes	characteristic
315	C₄	—	2	4	no	—
316	C₄	—	4	2	no	—
317	C₄	—	4	2	no	—
318	C₄	—	4	2	no	—
319	C₄	—	8	4	no	—
320	C₄	C₂²×C₇×C₄⋊C₄	1	1	yes	characteristic
321	C₄	C₂×C₄×C₇×D₄	1	1	yes	characteristic
322	C₄	C₇×#547	1	1	yes	characteristic
323	C₄	C₇×D₄×Q₈	1	1	yes	characteristic
324	C₂	—	2	1	no	—
325	C₂	—	2	1	no	—
326	C₂	—	4	4	no	—
327	C₂	C₇×D₄×C₄⋊C₄	1	1	yes	characteristic
328	C₂	C₇×#3290	1	2	yes	—

	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×#2912	C₂	semidirect
4. C₇×#1952	C₂	semidirect
5. C₇×#1860	C₂	semidirect
6. C₇×#1846	C₂	semidirect
7. C₇×#1815	C₂	semidirect
8. C₇×#1772	C₂	semidirect
12. C₇×#443	C₄	semidirect
32. #22690	C₇	direct (N × H)
40. C₇×#131	C₂×C₄	semidirect
56. C₇×#108	C₂×C₄	semidirect
66. C₇×#104	C₂×C₄	semidirect
71. C₄×C₇×C₈	C₂×C₄	semidirect
75. #2912	C₂×C₇	semidirect
76. #1952	C₂×C₇	semidirect
77. #1860	C₂×C₇	semidirect
78. #1846	C₂×C₇	semidirect
79. #1815	C₂×C₇	semidirect
80. #1772	C₂×C₇	semidirect
126. #443	C₄×C₇	semidirect
183. #131	C₂×C₄×C₇	semidirect
199. #108	C₂×C₄×C₇	semidirect
209. #104	C₂×C₄×C₇	semidirect
214. C₄×C₈	C₂×C₄×C₇	semidirect
308. C₇	#22690	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×#2912	2
C₇×#1952	2
C₇×#1860	2
C₇×#1846	2
C₇×#1815	2
C₇×#1772	2
#22690	7

Subgroups — C7×#22690 #13199906

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×#22690 #13199906