Subgroups — C₂×#11729104 #13508523

Order 1792.

558 conjugacy classes of subgroups.

The subgroup lattice has 286 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₇×#498	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#89472	C₂	1	1	yes	maximal, characteristic
3	#11729104	C₂	1	8	yes	maximal, direct product factor
4	C₂×#89283	C₂	1	2	yes	maximal
5	C₂×#89282	C₂	1	2	yes	maximal, complemented
6	C₂×#89258	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×C₇×C₄⋊Q₈	—	2	1	no	—
8	C₂×#6467	C₂²	1	1	yes	characteristic
9	C₂×#6464	—	2	1	no	—
10	C₇×#498	—	2	2	no	—
11	C₇×#498	C₂²	1	4	yes	complemented
12	C₂×C₇×#106	—	2	1	no	—
13	C₂×C₇×#106	C₂²	1	2	yes	—
14	C₂×C₄×C₇×C₈	C₂²	1	1	yes	characteristic
15	C₂×#6393	C₂²	1	1	yes	characteristic
16	C₂×#6386	C₂²	1	2	yes	—
17	C₂×#6379	—	2	1	no	—
18	#89472	—	2	2	no	—
19	#89472	C₂²	1	4	yes	—
20	#89283	C₂²	1	8	yes	—
21	#89282	C₂²	1	8	yes	complemented
22	#89258	C₂²	1	4	yes	complemented
23	C₂×#2230	—	7	1	no	maximal, Sylow
24	C₂²×C₇×Q₈	—	4	1	no	—
25	C₂×#1157	—	2	1	no	—
26	C₂²×C₇×C₈	D₄	1	1	yes	characteristic
27	C₂²×C₇×C₈	C₂³	1	1	yes	characteristic
28	C₇×C₄⋊Q₈	—	2	4	no	—
29	C₇×C₄⋊Q₈	—	4	2	no	—
30	C₂×C₇×C₄⋊C₄	—	2	2	no	—
31	C₂×C₇×C₄⋊C₄	—	4	1	no	—
32	C₂×C₄²×C₇	D₄	1	1	yes	characteristic
33	C₂×#1136	—	2	1	no	—
34	C₂×Dic₂₈	Q₈	1	1	yes	characteristic, complemented
35	#6467	—	2	1	no	—
36	#6467	—	2	1	no	—
37	#6467	Q₈	1	2	yes	complemented
38	#6467	C₂³	1	2	yes	—
39	C₂×D₅₆	Q₈	1	1	yes	characteristic, complemented
40	C₂×#1132	—	2	1	no	—
41	#6464	—	2	4	no	—
42	#6464	—	4	2	no	—
43	C₂×#1131	—	4	1	no	—
44	C₂×C₈×D₇	—	4	1	no	—
45	C₇×#106	—	2	2	no	—
46	C₇×#106	—	2	4	no	—
47	C₇×#106	C₂³	1	4	yes	—
48	C₄×C₇×C₈	D₄	1	2	yes	—
49	C₄×C₇×C₈	C₂³	1	2	yes	—
50	#6393	—	2	1	no	—
51	#6393	C₂³	1	2	yes	—
52	#6386	—	2	2	no	—
53	#6386	C₂³	1	4	yes	—
54	C₂×C₇⋊C₁₆	—	2	2	no	—
55	#6379	—	2	4	no	—
56	C₂×#498	D₇	1	1	yes	characteristic, complemented
57	C₂×#443	—	7	1	no	—
58	#2230	—	7	8	no	—
59	C₂×#365	—	7	2	no	—
60	C₂×#364	—	7	1	no	—
61	C₂×#360	—	7	2	no	—
62	C₂×C₇×Q₈	—	4	4	no	—
63	C₂×C₇×Q₈	—	8	4	no	—
64	C₂²×C₄×C₇	—	2	1	no	—
65	C₂²×C₄×C₇	—	4	2	no	—
66	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
67	C₂×C₂²⋊D₇	—	4	1	no	—
68	#1157	—	2	2	no	—
69	#1157	—	2	2	no	—
70	#1157	—	4	1	no	—
71	#1157	—	4	1	no	—
72	C₂×D₂₈	—	2	1	no	—
73	C₂×C₄×D₇	—	4	1	no	—
74	C₂×Dic₁₄	—	2	1	no	—
75	C₂×C₇×C₈	SD₁₆	1	1	yes	characteristic
76	C₂×C₇×C₈	SD₁₆	1	1	yes	characteristic
77	C₂×C₇×C₈	SD₁₆	1	2	yes	—
78	C₂×C₇×C₈	C₂×D₄	1	1	yes	characteristic
79	C₂×C₇×C₈	C₂×D₄	1	2	yes	—
80	C₂×C₇×C₈	C₂×Q₈	1	1	yes	characteristic
81	C₂×C₇×C₈	C₂×Q₈	1	1	yes	characteristic
82	C₂×C₇×C₈	#41	1	1	yes	characteristic
83	C₂×C₇×C₈	#41	1	1	yes	characteristic
84	C₂×C₇×C₈	C₂⁴	1	1	yes	characteristic
85	C₇×C₄⋊C₄	—	2	4	no	—
86	C₇×C₄⋊C₄	—	4	4	no	—
87	C₇×C₄⋊C₄	—	4	2	no	—
88	C₄²×C₇	—	2	1	no	—
89	C₄²×C₇	C₂×D₄	1	2	yes	—
90	#1136	—	2	2	no	—
91	#1136	—	4	1	no	—
92	C₂×C₇⋊C₈	—	4	1	no	—
93	Dic₂₈	—	2	1	no	—
94	Dic₂₈	C₂×Q₈	1	2	yes	complemented
95	D₅₆	—	2	1	no	—
96	D₅₆	C₂×Q₈	1	2	yes	complemented
97	#1132	—	2	2	no	—
98	#1132	—	2	2	no	—
99	#1131	—	4	4	no	—
100	C₈×D₇	—	4	4	no	—
101	C₇⋊C₁₆	—	2	4	no	—
102	C₂×#135	—	7	1	no	—
103	C₂×#131	—	14	1	no	—
104	C₂×C₄⋊Q₈	—	2	1	no	—
105	C₂×M₃₂	—	7	2	no	—
106	#498	—	2	2	no	—
107	#498	C₂×D₇	1	4	yes	complemented
108	#443	—	7	4	no	—
109	#443	—	14	2	no	—
110	C₂×#108	—	7	1	no	—
111	C₂×#106	—	2	1	no	—
112	C₂×#106	C₂×D₇	1	2	yes	—
113	C₂×#104	—	14	1	no	—
114	C₂×C₄×C₈	C₂×D₇	1	1	yes	characteristic
115	#365	—	7	8	no	—
116	#364	—	7	4	no	—
117	#360	—	7	8	no	—
118	C₂³×C₇	AGL(1,ℤ/8)	1	1	yes	characteristic
119	C₂²×D₇	—	4	1	no	—
120	C₇×Q₈	—	8	8	no	—
121	C₂×C₄×C₇	—	2	4	no	—
122	C₂×C₄×C₇	—	2	2	no	—
123	C₂×C₄×C₇	—	4	4	no	—
124	C₂×C₄×C₇	—	8	4	no	—
125	C₂×C₄×C₇	C₂²⋊Q₈	1	1	yes	characteristic
126	C₂×C₄×C₇	C₂²⋊Q₈	1	1	yes	characteristic
127	C₂×C₄×C₇	C₂×SD₁₆	1	1	yes	characteristic
128	C₂×C₄×C₇	C₂×SD₁₆	1	1	yes	characteristic
129	C₂×C₄×C₇	AGL(1,ℤ/8)	1	1	yes	characteristic
130	C₂×C₄×C₇	C₂²×D₄	1	1	yes	characteristic
131	C₂²⋊D₇	—	4	2	no	—
132	C₂²⋊D₇	—	4	2	no	—
133	C₂×Dic₇	—	4	1	no	—
134	D₂₈	—	2	2	no	—
135	D₂₈	—	4	1	no	—
136	C₄×D₇	—	4	2	no	—
137	C₄×D₇	—	4	2	no	—
138	Dic₁₄	—	2	2	no	—
139	Dic₁₄	—	4	1	no	—
140	C₇×C₈	C₂²⋊Q₈	1	1	yes	characteristic
141	C₇×C₈	C₂²⋊Q₈	1	1	yes	characteristic
142	C₇×C₈	C₂×SD₁₆	1	2	yes	—
143	C₇×C₈	C₂×SD₁₆	1	2	yes	—
144	C₇×C₈	C₂²×Q₈	1	1	yes	characteristic
145	C₇×C₈	C₂×#41	1	1	yes	characteristic
146	C₇⋊C₈	—	4	2	no	—
147	C₂×#41	—	14	1	no	—
148	C₂²×Q₈	—	4	1	no	—
149	#135	—	7	2	no	—
150	#135	—	7	2	no	—
151	#135	—	14	1	no	—
152	#135	—	14	1	no	—
153	C₂×Q₁₆	—	7	1	no	—
154	C₂×SD₁₆	—	14	1	no	—
155	C₂×D₈	—	7	1	no	—
156	#131	—	14	4	no	—
157	#131	—	28	2	no	—
158	C₂×M₁₆	—	14	1	no	—
159	C₂×M₁₆	—	28	1	no	—
160	C₂²×C₈	—	28	1	no	—
161	C₂²×C₈	C₂²⋊D₇	1	1	yes	characteristic
162	C₂²×C₈	C₂²×D₇	1	1	yes	characteristic
163	C₄⋊Q₈	—	2	4	no	—
164	C₄⋊Q₈	—	4	2	no	—
165	C₂×C₄⋊C₄	—	2	2	no	—
166	C₂×C₄⋊C₄	—	4	1	no	—
167	C₂×C₄²	C₂²⋊D₇	1	1	yes	characteristic
168	M₃₂	—	7	4	no	—
169	M₃₂	—	14	2	no	—
170	C₂×C₁₆	—	14	2	no	—
171	#108	—	7	2	no	—
172	#108	—	14	1	no	—
173	#106	—	2	4	no	—
174	#106	—	2	2	no	—
175	#106	C₂²×D₇	1	4	yes	—
176	#104	—	14	4	no	—
177	C₄×C₈	C₂²⋊D₇	1	2	yes	—
178	C₄×C₈	C₂²×D₇	1	2	yes	—
179	C₂²×C₇	—	2	2	no	—
180	C₂²×C₇	#476	1	1	yes	characteristic
181	C₂²×C₇	#476	1	1	yes	characteristic
182	C₂²×C₇	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
183	C₂×D₇	—	4	2	no	—
184	C₂×D₇	—	8	2	no	—
185	C₄×C₇	—	2	4	no	—
186	C₄×C₇	—	8	4	no	—
187	C₄×C₇	#476	1	1	yes	characteristic
188	C₄×C₇	#476	1	1	yes	characteristic
189	C₄×C₇	C₂×C₂²⋊Q₈	1	1	yes	characteristic
190	C₄×C₇	C₂²×SD₁₆	1	1	yes	characteristic
191	Dic₇	—	4	2	no	—
192	#41	—	14	2	no	—
193	#41	—	14	2	no	—
194	#41	—	28	1	no	—
195	#41	—	28	1	no	—
196	C₂×Q₈	—	4	4	no	—
197	C₂×Q₈	—	8	4	no	—
198	C₂×Q₈	—	14	1	no	—
199	C₂×D₄	—	14	1	no	—
200	C₂×D₄	—	28	1	no	—
201	C₂²×C₄	—	2	1	no	—
202	C₂²×C₄	—	4	2	no	—
203	C₂²×C₄	—	28	1	no	—
204	C₂²×C₄	C₂×C₂²⋊D₇	1	1	yes	characteristic
205	Q₁₆	—	7	2	no	—
206	Q₁₆	—	14	1	no	—
207	SD₁₆	—	14	2	no	—
208	SD₁₆	—	14	2	no	—
209	D₈	—	7	2	no	—
210	D₈	—	14	1	no	—
211	M₁₆	—	14	2	no	—
212	M₁₆	—	28	1	no	—
213	M₁₆	—	28	4	no	—
214	C₂×C₈	—	28	1	no	—
215	C₂×C₈	—	28	4	no	—
216	C₂×C₈	#1143	1	1	yes	characteristic
217	C₂×C₈	#1143	1	1	yes	characteristic
218	C₂×C₈	#1143	1	2	yes	—
219	C₂×C₈	Q₈×D₇	1	1	yes	characteristic
220	C₂×C₈	Q₈×D₇	1	1	yes	characteristic
221	C₂×C₈	#1161	1	1	yes	characteristic
222	C₂×C₈	#1161	1	1	yes	characteristic
223	C₂×C₈	C₂×C₂²⋊D₇	1	2	yes	—
224	C₂×C₈	C₂×C₂²⋊D₇	1	1	yes	characteristic
225	C₂×C₈	C₂³×D₇	1	1	yes	characteristic
226	C₄⋊C₄	—	2	4	no	—
227	C₄⋊C₄	—	4	4	no	—
228	C₄⋊C₄	—	4	2	no	—
229	C₄²	—	2	1	no	—
230	C₄²	C₂×C₂²⋊D₇	1	2	yes	—
231	C₁₆	—	14	4	no	—
232	C₂×C₇	—	2	1	no	—
233	C₂×C₇	—	2	1	no	—
234	C₂×C₇	#2230	1	2	yes	complemented
235	C₂×C₇	C₂×#476	1	1	yes	characteristic
236	D₇	—	8	2	no	—
237	C₂³	—	28	1	no	—
238	C₂³	#6495	1	1	yes	characteristic
239	Q₈	—	8	8	no	—
240	Q₈	—	14	2	no	—
241	Q₈	—	28	1	no	—
242	D₄	—	14	2	no	—
243	D₄	—	28	2	no	—
244	D₄	—	28	2	no	—
245	D₄	—	28	1	no	—
246	C₂×C₄	—	2	4	no	—
247	C₂×C₄	—	2	2	no	—
248	C₂×C₄	—	4	4	no	—
249	C₂×C₄	—	8	4	no	—
250	C₂×C₄	—	28	1	no	—
251	C₂×C₄	—	28	2	no	—
252	C₂×C₄	—	28	2	no	—
253	C₂×C₄	#6495	1	1	yes	characteristic
254	C₂×C₄	C₂×#1143	1	1	yes	characteristic
255	C₂×C₄	C₂×#1143	1	1	yes	characteristic
256	C₂×C₄	#6509	1	1	yes	characteristic
257	C₂×C₄	#6509	1	1	yes	characteristic
258	C₂×C₄	C₂²×C₂²⋊D₇	1	1	yes	characteristic
259	C₈	—	28	2	no	—
260	C₈	C₂×#1143	1	2	yes	—
261	C₈	C₂×#1143	1	2	yes	—
262	C₈	#6509	1	1	yes	characteristic
263	C₈	#6509	1	1	yes	characteristic
264	C₈	C₂×Q₈×D₇	1	1	yes	characteristic
265	C₈	C₂×#1161	1	1	yes	characteristic
266	C₇	C₂×#2230	1	1	yes	characteristic, complemented, Sylow
267	C₂²	—	2	2	no	—
268	C₂²	—	28	2	no	—
269	C₂²	—	56	2	no	—
270	C₂²	#89856	1	1	yes	characteristic
271	C₂²	#89856	1	1	yes	characteristic
272	C₂²	C₂×#6495	1	1	yes	characteristic
273	C₄	—	2	4	no	—
274	C₄	—	8	4	no	—
275	C₄	—	28	2	no	—
276	C₄	#89856	1	1	yes	characteristic
277	C₄	#89856	1	1	yes	characteristic
278	C₄	C₂²×#1143	1	1	yes	characteristic
279	C₄	C₂×#6509	1	1	yes	characteristic
280	C₂	—	2	1	no	—
281	C₂	—	2	1	no	—
282	C₂	—	56	2	no	—
283	C₂	#11729104	1	2	yes	direct product factor
284	C₂	C₂×#89856	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#89856
commutator, cosocle	C₂×C₇×C₈	C₂⁴
Frattini	C₂×C₈	C₂³×D₇
Fitting	C₂×C₇×#498	C₂
socle	C₂²×C₇	#476
prime radical	C₂×#6467	C₂²
hypercenter	C₂×#498	D₇
hypocenter	C₇	C₂×#2230
LCS	G ▷ C₂×C₇×C₈ ▷ C₄×C₇ ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×C₄×C₈ ◁ C₂×#498

Complements

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

N	H ≅ G/N	type
1. C₂×C₇×#498	C₂	semidirect
3. #11729104	C₂	direct (N × H)
5. C₂×#89282	C₂	semidirect
6. C₂×#89258	C₂	semidirect
11. C₇×#498	C₂²	semidirect
21. #89282	C₂²	semidirect
22. #89258	C₂²	semidirect
34. C₂×Dic₂₈	Q₈	semidirect
37. #6467	Q₈	semidirect
39. C₂×D₅₆	Q₈	semidirect
56. C₂×#498	D₇	semidirect
94. Dic₂₈	C₂×Q₈	semidirect
96. D₅₆	C₂×Q₈	semidirect
107. #498	C₂×D₇	semidirect
234. C₂×C₇	#2230	semidirect
266. C₇	C₂×#2230	semidirect
283. C₂	#11729104	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×#498	2
C₂×#89472	2
#11729104	2
C₂×#89283	2
C₂×#89282	2
C₂×#89258	2
C₂×#2230	7

Subgroups — C2×#11729104 #13508523

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#11729104 #13508523