Subgroups — C₂×#11726841 #13506452

Order 1792.

738 conjugacy classes of subgroups.

The subgroup lattice has 290 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×D₇×M₃₂	C₂	1	2	yes	maximal
2	C₂×#89480	C₂	1	1	yes	maximal, characteristic
3	C₂×C₇×#347	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×#89307	C₂	1	2	yes	maximal, complemented
5	C₂×#89261	C₂	1	1	yes	maximal, characteristic, complemented
6	#11726841	C₂	1	8	yes	maximal, direct product factor
7	C₂²×#1131	C₄	1	1	yes	characteristic
8	C₂²×C₈×D₇	C₂²	1	1	yes	characteristic
9	C₂×#6435	C₄	1	1	yes	characteristic
10	C₂×C₇×M₃₂	C₂²	1	2	yes	—
11	C₂×C₇×#97	C₂²	1	1	yes	characteristic
12	C₂×#6389	C₂²	1	1	yes	characteristic
13	C₂×C₈×Dic₇	C₄	1	1	yes	characteristic
14	C₂×#6386	C₂²	1	2	yes	—
15	C₂×C₄×C₇⋊C₈	C₄	1	1	yes	characteristic
16	D₇×M₃₂	C₄	1	8	yes	complemented
17	D₇×M₃₂	C₂²	1	8	yes	—
18	C₂×#6372	C₄	1	4	yes	complemented
19	C₂×D₇×C₁₆	C₄	1	4	yes	complemented
20	#89480	C₄	1	4	yes	—
21	#89480	C₂²	1	4	yes	—
22	C₇×#347	C₂²	1	4	yes	complemented
23	#89307	C₂²	1	8	yes	complemented
24	#89261	C₂²	1	4	yes	complemented
25	C₂×#2110	—	7	1	no	maximal, Sylow
26	C₂²×C₄×D₇	C₂×C₄	1	1	yes	characteristic
27	C₂²×C₇×C₈	C₂×C₄	1	1	yes	characteristic
28	C₂²×C₇×C₈	C₂³	1	1	yes	characteristic
29	C₂×C₄²×C₇	C₂×C₄	1	1	yes	characteristic
30	C₂×(C₂×D₇)⋊C₄	—	2	1	no	—
31	C₂×Dic₇⋊C₄	—	2	1	no	—
32	C₂×C₄×Dic₇	C₂×C₄	1	1	yes	characteristic
33	C₂²×C₇⋊C₈	C₂×C₄	1	1	yes	characteristic
34	C₂²×C₇⋊C₈	C₂×C₄	1	1	yes	characteristic
35	C₂×#1131	—	2	2	no	—
36	C₂×#1131	—	2	2	no	—
37	C₂×#1131	C₂×C₄	1	4	yes	—
38	C₂×C₈×D₇	C₂×C₄	1	2	yes	—
39	C₂×C₈×D₇	C₂×C₄	1	2	yes	—
40	C₂×C₈×D₇	C₂×C₄	1	2	yes	—
41	C₂×C₈×D₇	C₂×C₄	1	4	yes	—
42	C₂×C₈×D₇	C₂³	1	2	yes	—
43	#6435	—	2	2	no	—
44	#6435	C₂×C₄	1	4	yes	—
45	C₇×M₃₂	C₂×C₄	1	4	yes	—
46	C₇×M₃₂	C₂³	1	4	yes	—
47	C₂×C₇×C₁₆	C₂×C₄	1	4	yes	—
48	C₇×#97	C₂×C₄	1	2	yes	—
49	C₇×#97	C₂³	1	2	yes	—
50	#6389	C₂×C₄	1	2	yes	—
51	#6389	C₂³	1	2	yes	—
52	C₈×Dic₇	C₂×C₄	1	4	yes	—
53	#6386	C₂×C₄	1	4	yes	—
54	#6386	C₂³	1	4	yes	—
55	C₂×C₇⋊C₁₆	C₂×C₄	1	4	yes	—
56	C₄×C₇⋊C₈	C₂×C₄	1	4	yes	—
57	#6372	C₂×C₄	1	16	yes	complemented
58	D₇×C₁₆	C₂×C₄	1	16	yes	complemented
59	C₂²×M₃₂	—	7	2	no	—
60	C₂×#405	—	7	1	no	—
61	#2110	—	7	8	no	—
62	C₂×#347	—	7	2	no	—
63	C₂×#347	—	7	1	no	—
64	C₂×#347	D₇	1	1	yes	characteristic, complemented
65	C₂³×D₇	M₁₆	1	1	yes	characteristic
66	C₂²×C₄×C₇	—	2	1	no	—
67	C₂²×C₄×C₇	C₂²×C₄	1	1	yes	characteristic
68	C₂²×Dic₇	—	2	1	no	—
69	C₂²×Dic₇	M₁₆	1	1	yes	characteristic
70	C₂×C₄×D₇	C₄²	1	2	yes	—
71	C₂×C₄×D₇	C₄²	1	2	yes	—
72	C₂×C₄×D₇	M₁₆	1	2	yes	—
73	C₂×C₄×D₇	M₁₆	1	2	yes	—
74	C₂×C₄×D₇	M₁₆	1	2	yes	—
75	C₂×C₄×D₇	C₂²×C₄	1	2	yes	—
76	C₂×C₇×C₈	—	2	1	no	—
77	C₂×C₇×C₈	—	2	1	no	—
78	C₂×C₇×C₈	C₂²×C₄	1	2	yes	—
79	C₂×C₇×C₈	C₂²×C₄	1	1	yes	characteristic
80	C₂×C₇×C₈	C₂²×C₄	1	2	yes	—
81	C₂×C₇×C₈	C₂²×C₄	1	2	yes	—
82	C₂×C₇×C₈	C₂⁴	1	1	yes	characteristic
83	C₄²×C₇	—	2	1	no	—
84	C₄²×C₇	C₂²×C₄	1	2	yes	—
85	(C₂×D₇)⋊C₄	—	2	4	no	—
86	Dic₇⋊C₄	—	2	4	no	—
87	C₄×Dic₇	—	2	1	no	—
88	C₄×Dic₇	C₂²×C₄	1	2	yes	—
89	C₂×C₇⋊C₈	—	2	1	no	—
90	C₂×C₇⋊C₈	—	2	1	no	—
91	C₂×C₇⋊C₈	C₄²	1	2	yes	—
92	C₂×C₇⋊C₈	C₄²	1	1	yes	characteristic
93	C₂×C₇⋊C₈	C₄²	1	1	yes	characteristic
94	C₂×C₇⋊C₈	C₂²×C₄	1	2	yes	—
95	C₂×C₇⋊C₈	C₂²×C₄	1	2	yes	—
96	#1131	—	2	8	no	—
97	C₈×D₇	C₄²	1	4	yes	—
98	C₈×D₇	C₄²	1	4	yes	—
99	C₈×D₇	C₂²×C₄	1	4	yes	—
100	C₈×D₇	C₂²×C₄	1	4	yes	—
101	C₇×C₁₆	C₂²×C₄	1	8	yes	—
102	C₇⋊C₁₆	C₂²×C₄	1	8	yes	—
103	C₂²×M₁₆	—	7	1	no	—
104	C₂³×C₈	—	7	1	no	—
105	C₂×#117	—	7	1	no	—
106	C₂×M₃₂	—	7	2	no	—
107	C₂×M₃₂	—	7	8	no	—
108	C₂×M₃₂	—	7	8	no	—
109	C₂×M₃₂	—	7	4	no	—
110	C₂×M₃₂	C₂×D₇	1	2	yes	—
111	C₂²×C₁₆	—	7	4	no	—
112	#405	—	7	4	no	—
113	#405	—	7	4	no	—
114	C₂×#97	—	7	1	no	—
115	C₂×#97	C₂×D₇	1	1	yes	characteristic
116	C₂×C₄×C₈	—	7	1	no	—
117	C₂×C₄×C₈	—	7	1	no	—
118	#347	—	7	8	no	—
119	#347	—	7	4	no	—
120	#347	C₂×D₇	1	4	yes	complemented
121	C₂³×C₇	C₂×M₁₆	1	1	yes	characteristic
122	C₂²×D₇	—	4	2	no	—
123	C₂²×D₇	#97	1	2	yes	—
124	C₂²×D₇	#97	1	2	yes	—
125	C₂²×D₇	C₂×M₁₆	1	2	yes	—
126	C₂×C₄×C₇	—	2	2	no	—
127	C₂×C₄×C₇	—	4	2	no	—
128	C₂×C₄×C₇	C₂×C₄²	1	1	yes	characteristic
129	C₂×C₄×C₇	C₂×C₄²	1	1	yes	characteristic
130	C₂×C₄×C₇	C₂×M₁₆	1	1	yes	characteristic
131	C₂×C₄×C₇	C₂×M₁₆	1	1	yes	characteristic
132	C₂×C₄×C₇	C₂×M₁₆	1	1	yes	characteristic
133	C₂×C₄×C₇	C₂³×C₄	1	1	yes	characteristic
134	C₂×Dic₇	—	2	2	no	—
135	C₂×Dic₇	—	4	2	no	—
136	C₂×Dic₇	#97	1	2	yes	—
137	C₂×Dic₇	#97	1	2	yes	—
138	C₂×Dic₇	C₂×M₁₆	1	2	yes	—
139	C₄×D₇	#97	1	4	yes	—
140	C₄×D₇	#97	1	4	yes	—
141	C₄×D₇	C₂×C₄²	1	4	yes	—
142	C₄×D₇	C₂×M₁₆	1	4	yes	—
143	C₇×C₈	—	2	2	no	—
144	C₇×C₈	C₂×C₄²	1	2	yes	—
145	C₇×C₈	C₂³×C₄	1	2	yes	—
146	C₇⋊C₈	—	2	2	no	—
147	C₇⋊C₈	C₂×C₄²	1	2	yes	—
148	C₇⋊C₈	C₂×C₄²	1	2	yes	—
149	C₂³×C₄	—	7	1	no	—
150	C₂×M₁₆	—	7	4	no	—
151	C₂×M₁₆	—	14	2	no	—
152	C₂×M₁₆	—	14	2	no	—
153	C₂²×C₈	—	7	1	no	—
154	C₂²×C₈	—	7	2	no	—
155	C₂²×C₈	—	7	2	no	—
156	C₂²×C₈	—	7	4	no	—
157	C₂²×C₈	—	7	2	no	—
158	C₂²×C₈	—	7	2	no	—
159	C₂²×C₈	—	7	1	no	—
160	C₂²×C₈	C₄×D₇	1	1	yes	characteristic
161	C₂²×C₈	C₂²×D₇	1	1	yes	characteristic
162	#117	—	7	4	no	—
163	#117	—	14	2	no	—
164	C₂×C₄⋊C₄	—	14	1	no	—
165	C₂×C₂²⋊C₄	—	14	1	no	—
166	C₂×C₄²	—	7	1	no	—
167	C₂×C₄²	C₄×D₇	1	1	yes	characteristic
168	M₃₂	—	7	4	no	—
169	M₃₂	—	7	4	no	—
170	M₃₂	—	7	16	no	—
171	M₃₂	C₄×D₇	1	4	yes	—
172	M₃₂	C₂²×D₇	1	4	yes	—
173	C₂×C₁₆	—	7	4	no	—
174	C₂×C₁₆	—	7	16	no	—
175	C₂×C₁₆	C₄×D₇	1	4	yes	—
176	#97	—	7	2	no	—
177	#97	—	7	2	no	—
178	#97	C₄×D₇	1	2	yes	—
179	#97	C₂²×D₇	1	2	yes	—
180	C₄×C₈	—	7	4	no	—
181	C₄×C₈	—	7	4	no	—
182	C₂²×C₇	—	2	2	no	—
183	C₂²×C₇	C₂×#97	1	1	yes	characteristic
184	C₂²×C₇	C₂×#97	1	1	yes	characteristic
185	C₂²×C₇	C₂²×M₁₆	1	1	yes	characteristic
186	C₂×D₇	—	2	4	no	—
187	C₂×D₇	—	4	2	no	—
188	C₂×D₇	—	4	2	no	—
189	C₂×D₇	C₂×#97	1	4	yes	—
190	C₄×C₇	—	4	2	no	—
191	C₄×C₇	C₂×#97	1	1	yes	characteristic
192	C₄×C₇	C₂×#97	1	1	yes	characteristic
193	C₄×C₇	C₂²×C₄²	1	1	yes	characteristic
194	C₄×C₇	C₂²×M₁₆	1	1	yes	characteristic
195	Dic₇	—	4	2	no	—
196	Dic₇	C₂×#97	1	4	yes	—
197	C₂⁴	—	7	1	no	—
198	C₂²×C₄	—	2	1	no	—
199	C₂²×C₄	—	7	1	no	—
200	C₂²×C₄	—	7	2	no	—
201	C₂²×C₄	—	7	2	no	—
202	C₂²×C₄	—	7	2	no	—
203	C₂²×C₄	—	7	2	no	—
204	C₂²×C₄	—	7	2	no	—
205	C₂²×C₄	—	7	2	no	—
206	C₂²×C₄	—	14	1	no	—
207	C₂²×C₄	C₂×C₄×D₇	1	1	yes	characteristic
208	M₁₆	—	14	8	no	—
209	C₂×C₈	—	2	1	no	—
210	C₂×C₈	—	2	1	no	—
211	C₂×C₈	—	7	2	no	—
212	C₂×C₈	—	7	1	no	—
213	C₂×C₈	—	7	1	no	—
214	C₂×C₈	—	7	2	no	—
215	C₂×C₈	—	7	4	no	—
216	C₂×C₈	—	7	4	no	—
217	C₂×C₈	—	7	4	no	—
218	C₂×C₈	—	7	4	no	—
219	C₂×C₈	—	7	2	no	—
220	C₂×C₈	—	14	1	no	—
221	C₂×C₈	—	14	1	no	—
222	C₂×C₈	C₂×C₄×D₇	1	2	yes	—
223	C₂×C₈	C₂×C₄×D₇	1	1	yes	characteristic
224	C₂×C₈	C₂×C₄×D₇	1	2	yes	—
225	C₂×C₈	C₂×C₄×D₇	1	2	yes	—
226	C₂×C₈	C₂³×D₇	1	1	yes	characteristic
227	C₄⋊C₄	—	14	4	no	—
228	C₂²⋊C₄	—	14	4	no	—
229	C₄²	—	2	1	no	—
230	C₄²	—	7	2	no	—
231	C₄²	—	14	1	no	—
232	C₄²	C₂×C₄×D₇	1	2	yes	—
233	C₁₆	—	7	8	no	—
234	C₁₆	C₂×C₄×D₇	1	8	yes	—
235	C₂×C₇	—	2	1	no	—
236	C₂×C₇	—	2	1	no	—
237	C₂×C₇	#2110	1	2	yes	complemented
238	C₂×C₇	C₂²×#97	1	1	yes	characteristic
239	D₇	—	2	4	no	—
240	C₂³	—	7	2	no	—
241	C₂³	—	7	2	no	—
242	C₂³	—	7	2	no	—
243	C₂³	—	28	2	no	—
244	C₂³	D₇×M₁₆	1	1	yes	characteristic
245	C₂×C₄	—	2	2	no	—
246	C₂×C₄	—	4	2	no	—
247	C₂×C₄	—	7	2	no	—
248	C₂×C₄	—	7	2	no	—
249	C₂×C₄	—	7	2	no	—
250	C₂×C₄	—	7	4	no	—
251	C₂×C₄	—	7	4	no	—
252	C₂×C₄	—	7	4	no	—
253	C₂×C₄	—	7	4	no	—
254	C₂×C₄	—	14	2	no	—
255	C₂×C₄	—	28	2	no	—
256	C₂×C₄	C₄²×D₇	1	1	yes	characteristic
257	C₂×C₄	C₄²×D₇	1	1	yes	characteristic
258	C₂×C₄	D₇×M₁₆	1	1	yes	characteristic
259	C₂×C₄	D₇×M₁₆	1	1	yes	characteristic
260	C₂×C₄	D₇×M₁₆	1	1	yes	characteristic
261	C₂×C₄	C₂²×C₄×D₇	1	1	yes	characteristic
262	C₈	—	2	2	no	—
263	C₈	—	7	2	no	—
264	C₈	—	7	2	no	—
265	C₈	—	14	2	no	—
266	C₈	C₄²×D₇	1	2	yes	—
267	C₈	C₂²×C₄×D₇	1	2	yes	—
268	C₇	C₂×#2110	1	1	yes	characteristic, complemented, Sylow
269	C₂²	—	2	2	no	—
270	C₂²	—	7	4	no	—
271	C₂²	—	14	4	no	—
272	C₂²	—	28	2	no	—
273	C₂²	—	28	2	no	—
274	C₂²	D₇×#97	1	1	yes	characteristic
275	C₂²	D₇×#97	1	1	yes	characteristic
276	C₂²	C₂×D₇×M₁₆	1	1	yes	characteristic
277	C₄	—	4	2	no	—
278	C₄	—	7	4	no	—
279	C₄	—	28	2	no	—
280	C₄	D₇×#97	1	1	yes	characteristic
281	C₄	D₇×#97	1	1	yes	characteristic
282	C₄	C₂×C₄²×D₇	1	1	yes	characteristic
283	C₄	C₂×D₇×M₁₆	1	1	yes	characteristic
284	C₂	—	2	1	no	—
285	C₂	—	2	1	no	—
286	C₂	—	14	4	no	—
287	C₂	#11726841	1	2	yes	direct product factor
288	C₂	C₂×D₇×#97	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₄²×D₇
commutator	C₄×C₇	C₂²×C₄²
Frattini	C₂×C₈	C₂³×D₇
Fitting	C₂×C₇×#347	C₂
socle	C₂²×C₇	C₂×#97
prime radical	C₂³×D₇	M₁₆
cosocle	C₂×C₇×C₈	C₂⁴
hypercenter	C₂×#347	D₇
hypocenter	C₇	C₂×#2110
LCS	G ▷ C₄×C₇ ▷ C₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×#347

Complements

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

N	H ≅ G/N	type
3. C₂×C₇×#347	C₂	semidirect
4. C₂×#89307	C₂	semidirect
5. C₂×#89261	C₂	semidirect
6. #11726841	C₂	direct (N × H)
16. D₇×M₃₂	C₄	semidirect
18. C₂×#6372	C₄	semidirect
19. C₂×D₇×C₁₆	C₄	semidirect
22. C₇×#347	C₂²	semidirect
23. #89307	C₂²	semidirect
24. #89261	C₂²	semidirect
57. #6372	C₂×C₄	semidirect
58. D₇×C₁₆	C₂×C₄	semidirect
64. C₂×#347	D₇	semidirect
120. #347	C₂×D₇	semidirect
237. C₂×C₇	#2110	semidirect
268. C₇	C₂×#2110	semidirect
287. C₂	#11726841	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×D₇×M₃₂	2
C₂×#89480	2
C₂×C₇×#347	2
C₂×#89307	2
C₂×#89261	2
#11726841	2
C₂×#2110	7

Subgroups — C2×#11726841 #13506452

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#11726841 #13506452