Subgroups — C₂×#11727001 #13508530

Order 1792.

526 conjugacy classes of subgroups.

The subgroup lattice has 320 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#89968	C₂	1	1	yes	maximal, characteristic
2	C₂×#89630	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₇×#360	C₂	1	1	yes	maximal, characteristic
4	C₂×#89361	C₂	1	1	yes	maximal, characteristic
5	C₂×#89308	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#89306	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×#89282	C₂	1	1	yes	maximal, characteristic, complemented
8	#11727001	C₂	1	8	yes	maximal, direct product factor
9	C₂×C₇×#135	C₂²	1	1	yes	characteristic
10	C₂×#6511	—	2	1	no	—
11	C₂×#6453	—	2	1	no	—
12	C₂×C₇×M₃₂	C₂²	1	1	yes	characteristic
13	C₂×C₇×#106	C₂²	1	1	yes	characteristic
14	C₂×#6411	—	2	1	no	—
15	#89968	—	2	2	no	—
16	#89968	C₂²	1	4	yes	—
17	C₂×#6393	C₂²	1	1	yes	characteristic
18	C₂×#6391	C₂²	1	1	yes	characteristic
19	C₂×C₈×Dic₇	C₂²	1	1	yes	characteristic
20	C₂×#6386	C₂²	1	1	yes	characteristic
21	C₂×#6381	—	2	1	no	—
22	#89630	—	2	2	no	—
23	#89630	C₂²	1	4	yes	complemented
24	C₇×#360	C₂²	1	4	yes	—
25	#89361	C₂²	1	4	yes	—
26	#89308	C₂²	1	4	yes	complemented
27	#89306	C₂²	1	4	yes	complemented
28	#89282	C₂²	1	4	yes	complemented
29	C₂×#2231	—	7	1	no	maximal, Sylow
30	C₂×C₇×#41	—	2	1	no	—
31	C₇×#135	—	2	1	no	—
32	C₇×#135	—	2	1	no	—
33	C₇×#135	Q₈	1	2	yes	—
34	C₇×#135	C₂³	1	2	yes	—
35	C₂×C₇×Q₁₆	Q₈	1	1	yes	characteristic
36	C₂×C₇×SD₁₆	—	2	1	no	—
37	C₂×C₇×D₈	Q₈	1	1	yes	characteristic
38	C₂²×C₇×C₈	C₂³	1	1	yes	characteristic
39	C₂×C₇×C₄⋊C₄	—	2	1	no	—
40	#6511	—	2	4	no	—
41	#6511	—	4	2	no	—
42	C₂×C₄⋊Dic₇	—	2	1	no	—
43	C₂×C₄⋊Dic₇	—	4	1	no	—
44	C₂×Dic₇⋊C₄	—	4	1	no	—
45	C₂×C₄×Dic₇	D₄	1	1	yes	characteristic
46	C₂×#1136	—	2	1	no	—
47	C₂×#1136	—	4	1	no	—
48	C₂²×C₇⋊C₈	—	4	1	no	—
49	C₂²×C₇⋊C₈	D₄	1	1	yes	characteristic
50	#6453	—	2	4	no	—
51	#6453	—	4	2	no	—
52	C₇×M₃₂	—	2	1	no	—
53	C₇×M₃₂	C₂³	1	2	yes	—
54	C₂×C₇×C₁₆	—	2	1	no	—
55	C₇×#106	—	2	1	no	—
56	C₇×#106	C₂³	1	2	yes	—
57	#6411	—	2	4	no	—
58	#6393	—	2	1	no	—
59	#6393	C₂³	1	2	yes	—
60	#6391	—	2	1	no	—
61	#6391	C₂³	1	2	yes	—
62	C₈×Dic₇	D₄	1	2	yes	—
63	C₈×Dic₇	C₂³	1	2	yes	—
64	#6386	—	2	1	no	—
65	#6386	C₂³	1	2	yes	—
66	C₂×C₇⋊C₁₆	—	2	1	no	—
67	#6381	—	2	4	no	—
68	C₂×#499	—	7	1	no	—
69	C₂×#443	—	7	1	no	—
70	#2231	—	7	8	no	—
71	C₂×#365	—	7	1	no	—
72	C₂×#365	—	7	1	no	—
73	C₂×#364	—	7	1	no	—
74	C₂×#360	—	7	1	no	—
75	C₂×#360	D₇	1	1	yes	characteristic
76	C₇×#41	—	2	2	no	—
77	C₇×#41	—	2	2	no	—
78	C₇×#41	—	4	1	no	—
79	C₇×#41	—	4	1	no	—
80	C₂×C₇×Q₈	—	2	1	no	—
81	C₂×C₇×D₄	—	2	1	no	—
82	C₂×C₇×D₄	—	4	1	no	—
83	C₂²×C₄×C₇	—	4	1	no	—
84	C₂²×C₄×C₇	—	4	1	no	—
85	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
86	C₂²×Dic₇	—	2	1	no	—
87	C₂²×Dic₇	—	4	1	no	—
88	C₇×Q₁₆	—	2	1	no	—
89	C₇×Q₁₆	C₂×Q₈	1	2	yes	—
90	C₇×SD₁₆	—	2	2	no	—
91	C₇×SD₁₆	—	2	2	no	—
92	C₇×D₈	—	2	1	no	—
93	C₇×D₈	C₂×Q₈	1	2	yes	—
94	C₂×C₇×C₈	C₂×D₄	1	1	yes	characteristic
95	C₂×C₇×C₈	C₂×Q₈	1	1	yes	characteristic
96	C₂×C₇×C₈	C₂×Q₈	1	1	yes	characteristic
97	C₂×C₇×C₈	#41	1	1	yes	characteristic
98	C₂×C₇×C₈	#41	1	1	yes	characteristic
99	C₂×C₇×C₈	C₂⁴	1	1	yes	characteristic
100	C₇×C₄⋊C₄	—	2	2	no	—
101	C₇×C₄⋊C₄	—	4	1	no	—
102	C₄⋊Dic₇	—	2	2	no	—
103	C₄⋊Dic₇	—	4	4	no	—
104	C₄⋊Dic₇	—	4	1	no	—
105	Dic₇⋊C₄	—	4	4	no	—
106	C₄×Dic₇	—	2	1	no	—
107	C₄×Dic₇	C₂×D₄	1	2	yes	—
108	#1136	—	2	2	no	—
109	#1136	—	4	1	no	—
110	#1136	—	4	4	no	—
111	C₂×C₇⋊C₈	—	4	1	no	—
112	C₂×C₇⋊C₈	—	4	4	no	—
113	C₂×C₇⋊C₈	SD₁₆	1	1	yes	characteristic
114	C₂×C₇⋊C₈	SD₁₆	1	1	yes	characteristic
115	C₂×C₇⋊C₈	SD₁₆	1	2	yes	—
116	C₂×C₇⋊C₈	C₂×D₄	1	2	yes	—
117	C₇×C₁₆	—	2	2	no	—
118	C₇⋊C₁₆	—	2	2	no	—
119	C₂×#135	C₂×D₇	1	1	yes	characteristic
120	C₂×#131	—	14	1	no	—
121	C₂×#125	—	14	1	no	—
122	C₂×M₃₂	—	7	1	no	—
123	C₂×M₃₂	C₂×D₇	1	1	yes	characteristic
124	#499	—	7	4	no	—
125	#499	—	14	2	no	—
126	#443	—	7	4	no	—
127	#443	—	14	2	no	—
128	C₂×#108	—	7	1	no	—
129	C₂×#107	—	14	1	no	—
130	C₂×#106	—	7	1	no	—
131	C₂×#106	C₂×D₇	1	1	yes	characteristic
132	C₂×#104	—	14	1	no	—
133	C₂×C₄×C₈	—	7	1	no	—
134	#365	—	7	4	no	—
135	#365	—	7	4	no	—
136	#364	—	7	4	no	—
137	#360	—	7	4	no	—
138	#360	C₂×D₇	1	4	yes	—
139	C₂³×C₇	—	4	1	no	—
140	C₂³×C₇	AGL(1,ℤ/8)	1	1	yes	characteristic
141	C₇×Q₈	—	2	2	no	—
142	C₇×Q₈	—	4	1	no	—
143	C₇×D₄	—	2	2	no	—
144	C₇×D₄	—	4	2	no	—
145	C₇×D₄	—	4	2	no	—
146	C₇×D₄	—	4	1	no	—
147	C₂×C₄×C₇	—	4	1	no	—
148	C₂×C₄×C₇	—	4	2	no	—
149	C₂×C₄×C₇	—	4	2	no	—
150	C₂×C₄×C₇	—	4	2	no	—
151	C₂×C₄×C₇	—	8	2	no	—
152	C₂×C₄×C₇	C₂²⋊Q₈	1	1	yes	characteristic
153	C₂×C₄×C₇	C₂²⋊Q₈	1	1	yes	characteristic
154	C₂×C₄×C₇	C₂×SD₁₆	1	1	yes	characteristic
155	C₂×C₄×C₇	C₂×SD₁₆	1	1	yes	characteristic
156	C₂×C₄×C₇	AGL(1,ℤ/8)	1	1	yes	characteristic
157	C₂×C₄×C₇	C₂²×D₄	1	1	yes	characteristic
158	C₂×Dic₇	—	2	2	no	—
159	C₂×Dic₇	—	4	2	no	—
160	C₂×Dic₇	—	4	2	no	—
161	C₂×Dic₇	—	8	2	no	—
162	C₇×C₈	C₂²⋊Q₈	1	1	yes	characteristic
163	C₇×C₈	C₂²⋊Q₈	1	1	yes	characteristic
164	C₇×C₈	C₂²×Q₈	1	1	yes	characteristic
165	C₇×C₈	C₂×#41	1	1	yes	characteristic
166	C₇⋊C₈	—	4	2	no	—
167	C₇⋊C₈	C₂×SD₁₆	1	2	yes	—
168	C₇⋊C₈	C₂×SD₁₆	1	2	yes	—
169	C₂×#41	—	2	1	no	—
170	#135	—	2	1	no	—
171	#135	—	2	1	no	—
172	#135	Dic₁₄	1	2	yes	—
173	#135	C₂²×D₇	1	2	yes	—
174	C₂×Q₁₆	Dic₁₄	1	1	yes	characteristic
175	C₂×SD₁₆	—	2	1	no	—
176	C₂×D₈	Dic₁₄	1	1	yes	characteristic
177	#131	—	14	4	no	—
178	#131	—	28	2	no	—
179	C₂×M₁₆	—	14	1	no	—
180	C₂×M₁₆	—	28	1	no	—
181	C₂²×C₈	—	7	1	no	—
182	C₂²×C₈	—	28	1	no	—
183	C₂²×C₈	C₂²×D₇	1	1	yes	characteristic
184	#125	—	14	4	no	—
185	#125	—	28	2	no	—
186	C₂×C₄⋊C₄	—	2	1	no	—
187	C₂×C₄⋊C₄	—	14	1	no	—
188	C₂×C₄⋊C₄	—	28	1	no	—
189	C₂×C₄⋊C₄	—	28	1	no	—
190	C₂×C₄²	—	7	1	no	—
191	M₃₂	—	2	1	no	—
192	M₃₂	—	7	2	no	—
193	M₃₂	—	14	1	no	—
194	M₃₂	C₂²×D₇	1	2	yes	—
195	C₂×C₁₆	—	2	1	no	—
196	C₂×C₁₆	—	14	1	no	—
197	#108	—	7	2	no	—
198	#108	—	14	1	no	—
199	#107	—	14	4	no	—
200	#106	—	2	1	no	—
201	#106	—	7	2	no	—
202	#106	—	14	1	no	—
203	#106	C₂²×D₇	1	2	yes	—
204	#104	—	14	4	no	—
205	C₄×C₈	—	7	2	no	—
206	C₄×C₈	—	7	2	no	—
207	C₂²×C₇	—	2	2	no	—
208	C₂²×C₇	—	4	2	no	—
209	C₂²×C₇	—	8	2	no	—
210	C₂²×C₇	#476	1	1	yes	characteristic
211	C₂²×C₇	#476	1	1	yes	characteristic
212	C₂²×C₇	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
213	C₄×C₇	—	4	2	no	—
214	C₄×C₇	—	8	2	no	—
215	C₄×C₇	#476	1	1	yes	characteristic
216	C₄×C₇	#476	1	1	yes	characteristic
217	C₄×C₇	C₂×C₂²⋊Q₈	1	1	yes	characteristic
218	C₄×C₇	C₂²×SD₁₆	1	1	yes	characteristic
219	Dic₇	—	4	2	no	—
220	Dic₇	—	8	2	no	—
221	#41	—	2	2	no	—
222	#41	—	2	2	no	—
223	#41	—	4	1	no	—
224	#41	—	4	1	no	—
225	C₂×Q₈	—	2	1	no	—
226	C₂×D₄	—	2	1	no	—
227	C₂×D₄	—	4	1	no	—
228	C₂²×C₄	—	4	1	no	—
229	C₂²×C₄	—	4	1	no	—
230	C₂²×C₄	—	14	1	no	—
231	C₂²×C₄	—	28	1	no	—
232	C₂²×C₄	D₄×D₇	1	1	yes	characteristic
233	Q₁₆	—	2	1	no	—
234	Q₁₆	C₂×Dic₁₄	1	2	yes	—
235	SD₁₆	—	2	2	no	—
236	SD₁₆	—	2	2	no	—
237	D₈	—	2	1	no	—
238	D₈	C₂×Dic₁₄	1	2	yes	—
239	M₁₆	—	14	2	no	—
240	M₁₆	—	28	4	no	—
241	M₁₆	—	28	1	no	—
242	C₂×C₈	—	7	1	no	—
243	C₂×C₈	—	7	1	no	—
244	C₂×C₈	—	7	2	no	—
245	C₂×C₈	—	7	2	no	—
246	C₂×C₈	—	28	4	no	—
247	C₂×C₈	—	28	1	no	—
248	C₂×C₈	C₂×Dic₁₄	1	1	yes	characteristic
249	C₂×C₈	C₂×Dic₁₄	1	1	yes	characteristic
250	C₂×C₈	D₄×D₇	1	1	yes	characteristic
251	C₂×C₈	#1159	1	1	yes	characteristic
252	C₂×C₈	#1159	1	1	yes	characteristic
253	C₂×C₈	C₂³×D₇	1	1	yes	characteristic
254	C₄⋊C₄	—	2	2	no	—
255	C₄⋊C₄	—	4	1	no	—
256	C₄⋊C₄	—	14	2	no	—
257	C₄⋊C₄	—	28	4	no	—
258	C₄⋊C₄	—	28	4	no	—
259	C₄⋊C₄	—	28	1	no	—
260	C₄²	—	7	2	no	—
261	C₄²	—	14	1	no	—
262	C₁₆	—	2	2	no	—
263	C₁₆	—	14	2	no	—
264	C₂×C₇	—	2	1	no	—
265	C₂×C₇	—	2	1	no	—
266	C₂×C₇	—	8	2	no	—
267	C₂×C₇	#2231	1	2	yes	complemented
268	C₂×C₇	C₂×#476	1	1	yes	characteristic
269	C₂³	—	4	1	no	—
270	C₂³	#6474	1	1	yes	characteristic
271	Q₈	—	2	2	no	—
272	Q₈	—	4	1	no	—
273	D₄	—	2	2	no	—
274	D₄	—	4	2	no	—
275	D₄	—	4	2	no	—
276	D₄	—	4	1	no	—
277	C₂×C₄	—	4	1	no	—
278	C₂×C₄	—	4	2	no	—
279	C₂×C₄	—	4	2	no	—
280	C₂×C₄	—	4	2	no	—
281	C₂×C₄	—	8	2	no	—
282	C₂×C₄	—	14	2	no	—
283	C₂×C₄	—	28	2	no	—
284	C₂×C₄	—	28	2	no	—
285	C₂×C₄	—	56	2	no	—
286	C₂×C₄	#6441	1	1	yes	characteristic
287	C₂×C₄	#6441	1	1	yes	characteristic
288	C₂×C₄	#6474	1	1	yes	characteristic
289	C₂×C₄	D₇×SD₁₆	1	1	yes	characteristic
290	C₂×C₄	D₇×SD₁₆	1	1	yes	characteristic
291	C₂×C₄	C₂×D₄×D₇	1	1	yes	characteristic
292	C₈	—	7	2	no	—
293	C₈	—	7	2	no	—
294	C₈	—	28	2	no	—
295	C₈	#6441	1	1	yes	characteristic
296	C₈	#6441	1	1	yes	characteristic
297	C₈	C₂²×Dic₁₄	1	1	yes	characteristic
298	C₈	C₂×#1159	1	1	yes	characteristic
299	C₇	C₂×#2231	1	1	yes	characteristic, complemented, Sylow
300	C₂²	—	2	2	no	—
301	C₂²	—	4	2	no	—
302	C₂²	—	8	2	no	—
303	C₂²	#89533	1	1	yes	characteristic
304	C₂²	#89533	1	1	yes	characteristic
305	C₂²	C₂×#6474	1	1	yes	characteristic
306	C₄	—	4	2	no	—
307	C₄	—	8	2	no	—
308	C₄	—	28	2	no	—
309	C₄	—	56	2	no	—
310	C₄	#89533	1	1	yes	characteristic
311	C₄	#89533	1	1	yes	characteristic
312	C₄	C₂×#6441	1	1	yes	characteristic
313	C₄	C₂×D₇×SD₁₆	1	1	yes	characteristic
314	C₂	—	2	1	no	—
315	C₂	—	2	1	no	—
316	C₂	—	8	2	no	—
317	C₂	#11727001	1	2	yes	direct product factor
318	C₂	C₂×#89533	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#89533
commutator, cosocle	C₂×C₇×C₈	C₂⁴
Frattini	C₂×C₈	C₂³×D₇
Fitting	C₂×C₇×#360	C₂
socle	C₂²×C₇	#476
prime radical	C₂×C₇×#135	C₂²
hypercenter	C₂×#360	D₇
hypocenter	C₇	C₂×#2231
LCS	G ▷ C₂×C₇×C₈ ▷ C₄×C₇ ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂×#360

Complements

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

N	H ≅ G/N	type
2. C₂×#89630	C₂	semidirect
5. C₂×#89308	C₂	semidirect
6. C₂×#89306	C₂	semidirect
7. C₂×#89282	C₂	semidirect
8. #11727001	C₂	direct (N × H)
23. #89630	C₂²	semidirect
26. #89308	C₂²	semidirect
27. #89306	C₂²	semidirect
28. #89282	C₂²	semidirect
267. C₂×C₇	#2231	semidirect
299. C₇	C₂×#2231	semidirect
317. C₂	#11727001	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×#89968	2
C₂×#89630	2
C₂×C₇×#360	2
C₂×#89361	2
C₂×#89308	2
C₂×#89306	2
C₂×#89282	2
#11727001	2
C₂×#2231	7

Subgroups — C2×#11727001 #13508530

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#11727001 #13508530