Subgroups — D₄×#1136 #11739745

Order 896.

544 conjugacy classes of subgroups.

The subgroup lattice has 248 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×C₇×D₄	C₂	1	1	yes	maximal, characteristic
2	C₂²×#1136	C₂	1	2	yes	maximal, complemented
3	#89979	C₂	1	2	yes	maximal, complemented
4	#89784	C₂	1	2	yes	maximal, complemented
5	#89783	C₂	1	4	yes	maximal, complemented
6	D₄×C₇⋊C₈	C₂	1	2	yes	maximal, complemented
7	#89696	C₂	1	1	yes	maximal, characteristic, complemented
8	C₄×#1136	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂²×C₇×D₄	C₄	1	1	yes	characteristic
10	C₂³×C₄×C₇	C₂²	1	2	yes	—
11	C₄×C₇×D₄	C₄	1	2	yes	—
12	C₄×C₇×D₄	C₄	1	1	yes	characteristic
13	C₄×C₇×D₄	C₄	1	1	yes	characteristic
14	C₄×C₇×D₄	C₂²	1	1	yes	characteristic
15	C₄×C₇×D₄	C₂²	1	2	yes	—
16	C₄×C₇×D₄	C₂²	1	1	yes	characteristic
17	C₂×C₇×C₄⋊C₄	C₄	1	1	yes	characteristic
18	C₂×C₇×C₂²⋊C₄	C₄	1	2	yes	—
19	C₂×C₄²×C₇	C₂²	1	1	yes	characteristic
20	C₂×#1136	—	2	8	no	—
21	C₂×#1136	C₂²	1	1	yes	characteristic
22	C₂×#1136	C₂²	1	2	yes	—
23	C₂×#1136	C₂²	1	1	yes	characteristic
24	C₂×#1136	C₂²	1	4	yes	complemented
25	C₂²×C₇⋊C₈	C₂²	1	4	yes	complemented
26	C₂²⋊(C₇⋊C₈)	C₂²	1	4	yes	complemented
27	C₂²⋊(C₇⋊C₈)	C₂²	1	4	yes	complemented
28	C₄⋊(C₇⋊C₈)	C₂²	1	2	yes	complemented
29	C₄⋊(C₇⋊C₈)	C₂²	1	2	yes	complemented
30	#6377	C₂²	1	2	yes	complemented
31	C₄×C₇⋊C₈	C₂²	1	2	yes	complemented
32	D₄×M₁₆	—	7	1	no	maximal, Sylow
33	C₂⁴×C₇	C₂×C₄	1	2	yes	—
34	C₂×C₇×D₄	—	2	4	no	—
35	C₂×C₇×D₄	C₂×C₄	1	2	yes	—
36	C₂×C₇×D₄	C₂×C₄	1	1	yes	characteristic
37	C₂×C₇×D₄	C₂×C₄	1	1	yes	characteristic
38	C₂²×C₄×C₇	—	2	4	no	—
39	C₂²×C₄×C₇	—	2	4	no	—
40	C₂²×C₄×C₇	C₂×C₄	1	1	yes	characteristic
41	C₂²×C₄×C₇	C₂×C₄	1	2	yes	—
42	C₂²×C₄×C₇	C₂×C₄	1	2	yes	—
43	C₂²×C₄×C₇	C₂×C₄	1	2	yes	—
44	C₂²×C₄×C₇	C₂×C₄	1	1	yes	characteristic
45	C₂²×C₄×C₇	C₂³	1	2	yes	—
46	C₂²×C₄×C₇	C₂³	1	2	yes	—
47	C₂²×C₄×C₇	C₂³	1	1	yes	characteristic
48	C₇×C₄⋊C₄	C₂×C₄	1	2	yes	—
49	C₇×C₄⋊C₄	C₂×C₄	1	1	yes	characteristic
50	C₇×C₄⋊C₄	C₂×C₄	1	1	yes	characteristic
51	C₇×C₂²⋊C₄	C₂×C₄	1	2	yes	—
52	C₇×C₂²⋊C₄	C₂×C₄	1	2	yes	—
53	C₇×C₂²⋊C₄	C₂×C₄	1	2	yes	—
54	C₇×C₂²⋊C₄	C₂×C₄	1	2	yes	—
55	C₄²×C₇	C₂×C₄	1	1	yes	characteristic
56	C₄²×C₇	C₂×C₄	1	1	yes	characteristic
57	C₄²×C₇	C₂³	1	1	yes	characteristic
58	C₄²×C₇	C₂³	1	1	yes	characteristic
59	#1136	—	2	4	no	—
60	#1136	—	2	2	no	—
61	#1136	—	2	8	no	—
62	#1136	D₄	1	4	yes	direct product factor
63	C₂×C₇⋊C₈	—	2	8	no	—
64	C₂×C₇⋊C₈	C₂³	1	2	yes	—
65	C₂×C₇⋊C₈	C₂³	1	4	yes	—
66	C₂×C₇⋊C₈	C₂³	1	2	yes	—
67	C₂²×M₁₆	—	7	2	no	—
68	C₂×C₄×D₄	D₇	1	1	yes	characteristic
69	#436	—	7	2	no	—
70	#435	—	7	4	no	—
71	C₈×D₄	—	7	2	no	—
72	#423	—	7	1	no	—
73	#407	—	7	2	no	—
74	C₄×M₁₆	—	7	1	no	—
75	C₂³×C₇	—	2	4	no	—
76	C₂³×C₇	—	2	4	no	—
77	C₂³×C₇	—	4	2	no	—
78	C₂³×C₇	C₂²×C₄	1	2	yes	—
79	C₂³×C₇	C₂²×C₄	1	1	yes	characteristic
80	C₂³×C₇	C₂²×C₄	1	2	yes	—
81	C₇×D₄	—	2	4	no	—
82	C₇×D₄	—	2	2	no	—
83	C₇×D₄	M₁₆	1	4	yes	complemented
84	C₂×C₄×C₇	—	2	4	no	—
85	C₂×C₄×C₇	—	2	4	no	—
86	C₂×C₄×C₇	—	2	4	no	—
87	C₂×C₄×C₇	—	2	2	no	—
88	C₂×C₄×C₇	—	2	2	no	—
89	C₂×C₄×C₇	—	2	2	no	—
90	C₂×C₄×C₇	—	2	2	no	—
91	C₂×C₄×C₇	—	2	2	no	—
92	C₂×C₄×C₇	—	4	1	no	—
93	C₂×C₄×C₇	C₂²×C₄	1	1	yes	characteristic
94	C₂×C₄×C₇	C₂²×C₄	1	1	yes	characteristic
95	C₂×C₄×C₇	C₂²×C₄	1	1	yes	characteristic
96	C₂×C₄×C₇	C₂²×C₄	1	2	yes	—
97	C₂×C₄×C₇	C₂²×C₄	1	2	yes	—
98	C₂×C₄×C₇	C₂²×C₄	1	1	yes	characteristic
99	C₂×C₄×C₇	C₂²×C₄	1	1	yes	characteristic
100	C₂×C₄×C₇	C₂×D₄	1	2	yes	—
101	C₂×C₄×C₇	#41	1	2	yes	—
102	C₂×C₄×C₇	C₂⁴	1	1	yes	characteristic
103	C₇⋊C₈	—	2	4	no	—
104	C₇⋊C₈	—	2	2	no	—
105	C₇⋊C₈	C₂×D₄	1	4	yes	complemented
106	C₂²×D₄	Dic₇	1	1	yes	characteristic
107	C₂³×C₄	C₂×D₇	1	2	yes	—
108	C₂×M₁₆	—	7	1	no	—
109	C₂×M₁₆	—	7	2	no	—
110	C₂×M₁₆	—	7	4	no	—
111	C₂×M₁₆	—	7	1	no	—
112	C₂×M₁₆	—	14	8	no	—
113	C₂²×C₈	—	7	4	no	—
114	C₄×D₄	Dic₇	1	2	yes	—
115	C₄×D₄	Dic₇	1	1	yes	characteristic
116	C₄×D₄	Dic₇	1	1	yes	characteristic
117	C₄×D₄	C₂×D₇	1	1	yes	characteristic
118	C₄×D₄	C₂×D₇	1	2	yes	—
119	C₄×D₄	C₂×D₇	1	1	yes	characteristic
120	C₂×C₄⋊C₄	Dic₇	1	1	yes	characteristic
121	C₂×C₂²⋊C₄	Dic₇	1	2	yes	—
122	C₂×C₄²	C₂×D₇	1	1	yes	characteristic
123	C₄⋊C₈	—	7	2	no	—
124	C₄⋊C₈	—	7	2	no	—
125	C₂²⋊C₈	—	7	4	no	—
126	C₂²⋊C₈	—	7	4	no	—
127	#97	—	7	2	no	—
128	C₄×C₈	—	7	2	no	—
129	C₂²×C₇	—	2	2	no	—
130	C₂²×C₇	—	2	1	no	—
131	C₂²×C₇	—	2	1	no	—
132	C₂²×C₇	—	2	2	no	—
133	C₂²×C₇	—	2	2	no	—
134	C₂²×C₇	—	2	2	no	—
135	C₂²×C₇	—	2	2	no	—
136	C₂²×C₇	—	4	8	no	—
137	C₂²×C₇	C₄×D₄	1	2	yes	—
138	C₂²×C₇	C₂×M₁₆	1	4	yes	complemented
139	C₂²×C₇	C₂³×C₄	1	1	yes	characteristic
140	C₄×C₇	—	2	2	no	—
141	C₄×C₇	—	2	2	no	—
142	C₄×C₇	—	2	1	no	—
143	C₄×C₇	—	2	1	no	—
144	C₄×C₇	—	2	1	no	—
145	C₄×C₇	C₄×D₄	1	2	yes	—
146	C₄×C₇	C₂×M₁₆	1	2	yes	complemented
147	C₄×C₇	C₂²×D₄	1	1	yes	characteristic
148	C₄×C₇	C₂×#41	1	1	yes	characteristic
149	C₂⁴	C₂×Dic₇	1	2	yes	—
150	C₂×D₄	—	2	4	no	—
151	C₂×D₄	C₂×Dic₇	1	2	yes	—
152	C₂×D₄	C₂×Dic₇	1	1	yes	characteristic
153	C₂×D₄	C₂×Dic₇	1	1	yes	characteristic
154	C₂²×C₄	—	2	4	no	—
155	C₂²×C₄	—	2	4	no	—
156	C₂²×C₄	C₂×Dic₇	1	1	yes	characteristic
157	C₂²×C₄	C₂×Dic₇	1	2	yes	—
158	C₂²×C₄	C₂×Dic₇	1	2	yes	—
159	C₂²×C₄	C₂×Dic₇	1	2	yes	—
160	C₂²×C₄	C₂×Dic₇	1	1	yes	characteristic
161	C₂²×C₄	C₂²×D₇	1	2	yes	—
162	C₂²×C₄	C₂²×D₇	1	2	yes	—
163	C₂²×C₄	C₂²×D₇	1	1	yes	characteristic
164	M₁₆	—	7	4	no	—
165	M₁₆	—	14	8	no	—
166	M₁₆	—	14	4	no	—
167	M₁₆	—	14	2	no	—
168	C₂×C₈	—	7	2	no	—
169	C₂×C₈	—	7	2	no	—
170	C₂×C₈	—	7	4	no	—
171	C₂×C₈	—	14	8	no	—
172	C₄⋊C₄	C₂×Dic₇	1	2	yes	—
173	C₄⋊C₄	C₂×Dic₇	1	1	yes	characteristic
174	C₄⋊C₄	C₂×Dic₇	1	1	yes	characteristic
175	C₂²⋊C₄	C₂×Dic₇	1	2	yes	—
176	C₂²⋊C₄	C₂×Dic₇	1	2	yes	—
177	C₂²⋊C₄	C₂×Dic₇	1	2	yes	—
178	C₂²⋊C₄	C₂×Dic₇	1	2	yes	—
179	C₄²	C₂×Dic₇	1	1	yes	characteristic
180	C₄²	C₂×Dic₇	1	1	yes	characteristic
181	C₄²	C₂²×D₇	1	1	yes	characteristic
182	C₄²	C₂²×D₇	1	1	yes	characteristic
183	C₂×C₇	—	2	2	no	—
184	C₂×C₇	—	2	4	no	—
185	C₂×C₇	—	4	2	no	—
186	C₂×C₇	C₂×C₄×D₄	1	1	yes	characteristic
187	C₂×C₇	C₂²×M₁₆	1	1	yes	characteristic
188	C₂×C₇	#568	1	1	yes	characteristic
189	C₂³	—	2	4	no	—
190	C₂³	—	2	4	no	—
191	C₂³	—	4	2	no	—
192	C₂³	C₂²×Dic₇	1	2	yes	—
193	C₂³	C₂²×Dic₇	1	2	yes	—
194	C₂³	C₂²×Dic₇	1	1	yes	characteristic
195	D₄	—	2	4	no	—
196	D₄	—	2	2	no	—
197	D₄	#1136	1	4	yes	direct product factor
198	C₂×C₄	—	2	4	no	—
199	C₂×C₄	—	2	2	no	—
200	C₂×C₄	—	2	4	no	—
201	C₂×C₄	—	2	4	no	—
202	C₂×C₄	—	2	2	no	—
203	C₂×C₄	—	2	2	no	—
204	C₂×C₄	—	2	2	no	—
205	C₂×C₄	—	2	2	no	—
206	C₂×C₄	—	4	1	no	—
207	C₂×C₄	D₄×D₇	1	2	yes	—
208	C₂×C₄	#1159	1	2	yes	—
209	C₂×C₄	C₂²×Dic₇	1	1	yes	characteristic
210	C₂×C₄	C₂²×Dic₇	1	1	yes	characteristic
211	C₂×C₄	C₂²×Dic₇	1	1	yes	characteristic
212	C₂×C₄	C₂²×Dic₇	1	2	yes	—
213	C₂×C₄	C₂²×Dic₇	1	2	yes	—
214	C₂×C₄	C₂²×Dic₇	1	1	yes	characteristic
215	C₂×C₄	C₂²×Dic₇	1	1	yes	characteristic
216	C₂×C₄	C₂³×D₇	1	1	yes	characteristic
217	C₈	—	7	4	no	—
218	C₈	—	14	4	no	—
219	C₈	—	14	2	no	—
220	C₇	D₄×M₁₆	1	1	yes	characteristic, complemented, Sylow
221	C₂²	—	2	1	no	—
222	C₂²	—	2	2	no	—
223	C₂²	—	2	1	no	—
224	C₂²	—	2	2	no	—
225	C₂²	—	2	2	no	—
226	C₂²	—	2	2	no	—
227	C₂²	—	2	2	no	—
228	C₂²	—	4	8	no	—
229	C₂²	C₂×#1136	1	4	yes	complemented
230	C₂²	D₄×Dic₇	1	2	yes	—
231	C₂²	C₂³×Dic₇	1	1	yes	characteristic
232	C₄	—	2	1	no	—
233	C₄	—	2	2	no	—
234	C₄	—	2	2	no	—
235	C₄	—	2	1	no	—
236	C₄	—	2	1	no	—
237	C₄	C₂×#1136	1	2	yes	complemented
238	C₄	D₄×Dic₇	1	2	yes	—
239	C₄	C₂×D₄×D₇	1	1	yes	characteristic
240	C₄	C₂×#1159	1	1	yes	characteristic
241	C₂	—	2	2	no	—
242	C₂	—	2	4	no	—
243	C₂	—	4	2	no	—
244	C₂	C₂²×#1136	1	1	yes	characteristic
245	C₂	C₂×D₄×Dic₇	1	1	yes	characteristic
246	C₂	#90510	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₇
commutator, socle	C₂²×C₇	C₂³×C₄
Fitting	C₂×C₄×C₇×D₄	C₂
prime radical	C₂²×C₇×D₄	C₄
cosocle	C₂×C₄×C₇	C₂⁴
hypercenter	C₂×C₄×D₄	D₇
hypocenter	C₇	D₄×M₁₆
LCS	G ▷ C₂²×C₇ ▷ C₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄×D₄

Complements

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

N	H ≅ G/N	type
2. C₂²×#1136	C₂	semidirect
3. #89979	C₂	semidirect
4. #89784	C₂	semidirect
5. #89783	C₂	semidirect
6. D₄×C₇⋊C₈	C₂	semidirect
7. #89696	C₂	semidirect
8. C₄×#1136	C₂	semidirect
24. C₂×#1136	C₂²	semidirect
25. C₂²×C₇⋊C₈	C₂²	semidirect
26. C₂²⋊(C₇⋊C₈)	C₂²	semidirect
27. C₂²⋊(C₇⋊C₈)	C₂²	semidirect
28. C₄⋊(C₇⋊C₈)	C₂²	semidirect
29. C₄⋊(C₇⋊C₈)	C₂²	semidirect
30. #6377	C₂²	semidirect
31. C₄×C₇⋊C₈	C₂²	semidirect
62. #1136	D₄	direct (N × H)
83. C₇×D₄	M₁₆	semidirect
105. C₇⋊C₈	C₂×D₄	semidirect
138. C₂²×C₇	C₂×M₁₆	semidirect
146. C₄×C₇	C₂×M₁₆	semidirect
197. D₄	#1136	direct (N × H)
220. C₇	D₄×M₁₆	semidirect
229. C₂²	C₂×#1136	semidirect
237. C₄	C₂×#1136	semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂×C₄×C₇×D₄	2
C₂²×#1136	2
#89979	2
#89784	2
#89783	2
D₄×C₇⋊C₈	2
#89696	2
C₄×#1136	2
D₄×M₁₆	7

Subgroups — D4×#1136 #11739745

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — D₄×#1136 #11739745