Subgroups — D₄×C₄⋊C₄×C₂₂₇ #44396304

Order 29056.

776 conjugacy classes of subgroups.

The subgroup lattice has 236 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×D₄×C₂₂₇	C₂	1	2	yes	maximal
2	C₂×C₄×D₄×C₂₂₇	C₂	1	1	yes	maximal, characteristic
3	C₂²×C₄⋊C₄×C₂₂₇	C₂	1	2	yes	maximal, complemented
4	#389×C₂₂₇	C₂	1	2	yes	maximal, complemented
5	#385×C₂₂₇	C₂	1	4	yes	maximal, complemented
6	#384×C₂₂₇	C₂	1	1	yes	maximal, characteristic, complemented
7	#380×C₂₂₇	C₂	1	2	yes	maximal, complemented
8	C₄×C₄⋊C₄×C₂₂₇	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂²×D₄×C₂₂₇	C₂²	1	1	yes	characteristic
10	C₂³×C₄×C₂₂₇	C₂²	1	4	yes	—
11	C₂³×C₄×C₂₂₇	C₂²	1	2	yes	—
12	C₄×D₄×C₂₂₇	—	2	8	no	—
13	C₄×D₄×C₂₂₇	C₄	1	8	yes	complemented
14	C₂×C₄⋊C₄×C₂₂₇	—	2	8	no	—
15	C₂×C₄⋊C₄×C₂₂₇	C₂²	1	2	yes	—
16	C₂×C₄⋊C₄×C₂₂₇	C₂²	1	4	yes	—
17	C₂×C₄⋊C₄×C₂₂₇	C₂²	1	2	yes	—
18	C₂×C₄⋊C₄×C₂₂₇	C₂²	1	1	yes	characteristic
19	C₂×C₄⋊C₄×C₂₂₇	C₂²	1	2	yes	—
20	C₂×C₄⋊C₄×C₂₂₇	C₂²	1	1	yes	characteristic
21	C₂×C₄⋊C₄×C₂₂₇	C₂²	1	1	yes	characteristic
22	C₂×C₂²⋊C₄×C₂₂₇	C₂²	1	4	yes	—
23	C₂×C₂²⋊C₄×C₂₂₇	C₂²	1	2	yes	—
24	C₂×C₄²×C₂₂₇	C₂²	1	2	yes	—
25	C₂×C₄²×C₂₂₇	C₂²	1	1	yes	characteristic
26	#95×C₂₂₇	C₂²	1	2	yes	—
27	#95×C₂₂₇	C₂²	1	4	yes	—
28	C₂⁴×C₂₂₇	C₂³	1	2	yes	—
29	C₂×D₄×C₂₂₇	C₂×C₄	1	4	yes	—
30	C₂×D₄×C₂₂₇	D₄	1	4	yes	—
31	C₂×D₄×C₂₂₇	Q₈	1	4	yes	—
32	C₂²×C₄×C₂₂₇	—	2	4	no	—
33	C₂²×C₄×C₂₂₇	—	2	4	no	—
34	C₂²×C₄×C₂₂₇	—	2	8	no	—
35	C₂²×C₄×C₂₂₇	—	2	8	no	—
36	C₂²×C₄×C₂₂₇	—	4	4	no	—
37	C₂²×C₄×C₂₂₇	C₂×C₄	1	8	yes	complemented
38	C₂²×C₄×C₂₂₇	C₂³	1	2	yes	—
39	C₂²×C₄×C₂₂₇	C₂³	1	4	yes	—
40	C₂²×C₄×C₂₂₇	C₂³	1	1	yes	characteristic
41	C₂²×C₄×C₂₂₇	C₂³	1	2	yes	—
42	C₂²×C₄×C₂₂₇	C₂³	1	2	yes	—
43	C₂²×C₄×C₂₂₇	C₂³	1	1	yes	characteristic
44	C₂²×C₄×C₂₂₇	C₂³	1	1	yes	characteristic
45	C₄⋊C₄×C₂₂₇	—	2	8	no	—
46	C₄⋊C₄×C₂₂₇	—	2	4	no	—
47	C₄⋊C₄×C₂₂₇	—	2	4	no	—
48	C₄⋊C₄×C₂₂₇	—	2	2	no	—
49	C₄⋊C₄×C₂₂₇	—	4	2	no	—
50	C₄⋊C₄×C₂₂₇	C₂×C₄	1	4	yes	complemented
51	C₄⋊C₄×C₂₂₇	D₄	1	4	yes	direct product factor
52	C₂²⋊C₄×C₂₂₇	—	2	8	no	—
53	C₂²⋊C₄×C₂₂₇	C₂×C₄	1	8	yes	complemented
54	C₄²×C₂₂₇	—	2	4	no	—
55	C₄²×C₂₂₇	C₂×C₄	1	4	yes	complemented
56	C₂³×C₂₂₇	—	2	2	no	—
57	C₂³×C₂₂₇	—	2	2	no	—
58	C₂³×C₂₂₇	—	2	2	no	—
59	C₂³×C₂₂₇	—	2	2	no	—
60	C₂³×C₂₂₇	C₂²×C₄	1	4	yes	—
61	C₂³×C₂₂₇	C₂×D₄	1	4	yes	—
62	C₂³×C₂₂₇	C₂×Q₈	1	4	yes	—
63	C₂³×C₂₂₇	C₂⁴	1	1	yes	characteristic
64	D₄×C₂₂₇	C₄⋊C₄	1	16	yes	direct product factor
65	C₂×C₄×C₂₂₇	—	2	2	no	—
66	C₂×C₄×C₂₂₇	—	2	2	no	—
67	C₂×C₄×C₂₂₇	—	2	2	no	—
68	C₂×C₄×C₂₂₇	—	2	1	no	—
69	C₂×C₄×C₂₂₇	—	2	2	no	—
70	C₂×C₄×C₂₂₇	—	2	1	no	—
71	C₂×C₄×C₂₂₇	—	2	16	no	—
72	C₂×C₄×C₂₂₇	—	2	2	no	—
73	C₂×C₄×C₂₂₇	—	2	4	no	—
74	C₂×C₄×C₂₂₇	—	2	4	no	—
75	C₂×C₄×C₂₂₇	—	2	2	no	—
76	C₂×C₄×C₂₂₇	—	2	4	no	—
77	C₂×C₄×C₂₂₇	—	2	2	no	—
78	C₂×C₄×C₂₂₇	—	4	16	no	—
79	C₂×C₄×C₂₂₇	C₂²×C₄	1	2	yes	—
80	C₂×C₄×C₂₂₇	C₂²×C₄	1	4	yes	—
81	C₂×C₄×C₂₂₇	C₂²×C₄	1	2	yes	—
82	C₂×C₄×C₂₂₇	C₂²×C₄	1	2	yes	—
83	C₂×C₄×C₂₂₇	C₂×D₄	1	4	yes	—
84	C₂×C₄×C₂₂₇	C₂×D₄	1	2	yes	—
85	C₂×C₄×C₂₂₇	C₂×D₄	1	2	yes	—
86	C₂×C₄×C₂₂₇	C₂×Q₈	1	2	yes	—
87	C₂×C₄×C₂₂₇	#41	1	2	yes	—
88	C₂²×C₂₂₇	—	2	4	no	—
89	C₂²×C₂₂₇	—	2	4	no	—
90	C₂²×C₂₂₇	—	2	4	no	—
91	C₂²×C₂₂₇	—	2	4	no	—
92	C₂²×C₂₂₇	—	2	4	no	—
93	C₂²×C₂₂₇	—	2	4	no	—
94	C₂²×C₂₂₇	C₂×C₄⋊C₄	1	8	yes	complemented
95	C₂²×C₂₂₇	C₂³×C₄	1	1	yes	characteristic
96	C₂²×C₂₂₇	C₂²×D₄	1	1	yes	characteristic
97	C₂²×C₂₂₇	C₂²×D₄	1	1	yes	characteristic
98	C₂²×C₂₂₇	C₂²×Q₈	1	1	yes	characteristic
99	C₂²×C₂₂₇	C₂×#41	1	1	yes	characteristic
100	C₂²×C₂₂₇	H(5,𝔽₂)	1	1	yes	characteristic
101	C₂²×C₂₂₇	2₋¹⁺⁴	1	1	yes	characteristic
102	C₄×C₂₂₇	—	2	4	no	—
103	C₄×C₂₂₇	—	2	2	no	—
104	C₄×C₂₂₇	—	2	4	no	—
105	C₄×C₂₂₇	—	4	4	no	—
106	C₄×C₂₂₇	—	4	2	no	—
107	C₄×C₂₂₇	C₂×C₄⋊C₄	1	4	yes	complemented
108	C₄×C₂₂₇	C₄×D₄	1	4	yes	complemented
109	C₂×C₂₂₇	—	2	8	no	—
110	C₂×C₂₂₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
111	C₂×C₂₂₇	C₂×C₄×D₄	1	1	yes	characteristic
112	C₂×C₂₂₇	#520	1	1	yes	characteristic
113	C₂×C₂₂₇	D₄²	1	1	yes	characteristic
114	C₂×C₂₂₇	#547	1	1	yes	characteristic
115	C₂×C₂₂₇	D₄×Q₈	1	1	yes	characteristic
116	C₂×C₂₂₇	#554	1	1	yes	characteristic
117	C₂₂₇	D₄×C₄⋊C₄	1	1	yes	characteristic, direct product factor, Sylow
118	D₄×C₄⋊C₄	C₂₂₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
119	C₂×C₄×D₄	C₂×C₂₂₇	1	2	yes	—
120	C₂×C₄×D₄	C₂×C₂₂₇	1	1	yes	characteristic
121	C₂²×C₄⋊C₄	C₂×C₂₂₇	1	2	yes	complemented
122	#389	C₂×C₂₂₇	1	2	yes	complemented
123	#385	C₂×C₂₂₇	1	4	yes	complemented
124	#384	C₂×C₂₂₇	1	1	yes	characteristic, complemented
125	#380	C₂×C₂₂₇	1	2	yes	complemented
126	C₄×C₄⋊C₄	C₂×C₂₂₇	1	1	yes	characteristic, complemented
127	C₂²×D₄	C₂²×C₂₂₇	1	1	yes	characteristic
128	C₂³×C₄	C₂²×C₂₂₇	1	4	yes	—
129	C₂³×C₄	C₂²×C₂₂₇	1	2	yes	—
130	C₄×D₄	—	2	8	no	—
131	C₄×D₄	C₄×C₂₂₇	1	8	yes	complemented
132	C₂×C₄⋊C₄	—	2	8	no	—
133	C₂×C₄⋊C₄	C₂²×C₂₂₇	1	2	yes	—
134	C₂×C₄⋊C₄	C₂²×C₂₂₇	1	4	yes	—
135	C₂×C₄⋊C₄	C₂²×C₂₂₇	1	2	yes	—
136	C₂×C₄⋊C₄	C₂²×C₂₂₇	1	1	yes	characteristic
137	C₂×C₄⋊C₄	C₂²×C₂₂₇	1	2	yes	—
138	C₂×C₄⋊C₄	C₂²×C₂₂₇	1	1	yes	characteristic
139	C₂×C₄⋊C₄	C₂²×C₂₂₇	1	1	yes	characteristic
140	C₂×C₂²⋊C₄	C₂²×C₂₂₇	1	4	yes	—
141	C₂×C₂²⋊C₄	C₂²×C₂₂₇	1	2	yes	—
142	C₂×C₄²	C₂²×C₂₂₇	1	2	yes	—
143	C₂×C₄²	C₂²×C₂₂₇	1	1	yes	characteristic
144	#95	C₂²×C₂₂₇	1	2	yes	—
145	#95	C₂²×C₂₂₇	1	4	yes	—
146	C₂⁴	C₂³×C₂₂₇	1	2	yes	—
147	C₂×D₄	C₂×C₄×C₂₂₇	1	4	yes	—
148	C₂×D₄	D₄×C₂₂₇	1	4	yes	—
149	C₂×D₄	Q₈×C₂₂₇	1	4	yes	—
150	C₂²×C₄	—	2	4	no	—
151	C₂²×C₄	—	2	4	no	—
152	C₂²×C₄	—	2	8	no	—
153	C₂²×C₄	—	2	8	no	—
154	C₂²×C₄	—	4	4	no	—
155	C₂²×C₄	C₂×C₄×C₂₂₇	1	8	yes	complemented
156	C₂²×C₄	C₂³×C₂₂₇	1	2	yes	—
157	C₂²×C₄	C₂³×C₂₂₇	1	4	yes	—
158	C₂²×C₄	C₂³×C₂₂₇	1	1	yes	characteristic
159	C₂²×C₄	C₂³×C₂₂₇	1	2	yes	—
160	C₂²×C₄	C₂³×C₂₂₇	1	2	yes	—
161	C₂²×C₄	C₂³×C₂₂₇	1	1	yes	characteristic
162	C₂²×C₄	C₂³×C₂₂₇	1	1	yes	characteristic
163	C₄⋊C₄	—	2	8	no	—
164	C₄⋊C₄	—	2	4	no	—
165	C₄⋊C₄	—	2	4	no	—
166	C₄⋊C₄	—	2	2	no	—
167	C₄⋊C₄	—	4	2	no	—
168	C₄⋊C₄	C₂×C₄×C₂₂₇	1	4	yes	complemented
169	C₄⋊C₄	D₄×C₂₂₇	1	4	yes	direct product factor
170	C₂²⋊C₄	—	2	8	no	—
171	C₂²⋊C₄	C₂×C₄×C₂₂₇	1	8	yes	complemented
172	C₄²	—	2	4	no	—
173	C₄²	C₂×C₄×C₂₂₇	1	4	yes	complemented
174	C₂³	—	2	2	no	—
175	C₂³	—	2	2	no	—
176	C₂³	—	2	2	no	—
177	C₂³	—	2	2	no	—
178	C₂³	C₂²×C₄×C₂₂₇	1	4	yes	—
179	C₂³	C₂×D₄×C₂₂₇	1	4	yes	—
180	C₂³	C₂×Q₈×C₂₂₇	1	4	yes	—
181	C₂³	C₂⁴×C₂₂₇	1	1	yes	characteristic
182	D₄	C₄⋊C₄×C₂₂₇	1	16	yes	direct product factor
183	C₂×C₄	—	2	2	no	—
184	C₂×C₄	—	2	2	no	—
185	C₂×C₄	—	2	2	no	—
186	C₂×C₄	—	2	1	no	—
187	C₂×C₄	—	2	2	no	—
188	C₂×C₄	—	2	1	no	—
189	C₂×C₄	—	2	16	no	—
190	C₂×C₄	—	2	2	no	—
191	C₂×C₄	—	2	4	no	—
192	C₂×C₄	—	2	4	no	—
193	C₂×C₄	—	2	2	no	—
194	C₂×C₄	—	2	4	no	—
195	C₂×C₄	—	2	2	no	—
196	C₂×C₄	—	4	16	no	—
197	C₂×C₄	C₂²×C₄×C₂₂₇	1	2	yes	—
198	C₂×C₄	C₂²×C₄×C₂₂₇	1	4	yes	—
199	C₂×C₄	C₂²×C₄×C₂₂₇	1	2	yes	—
200	C₂×C₄	C₂²×C₄×C₂₂₇	1	2	yes	—
201	C₂×C₄	C₂×D₄×C₂₂₇	1	4	yes	—
202	C₂×C₄	C₂×D₄×C₂₂₇	1	2	yes	—
203	C₂×C₄	C₂×D₄×C₂₂₇	1	2	yes	—
204	C₂×C₄	C₂×Q₈×C₂₂₇	1	2	yes	—
205	C₂×C₄	#41×C₂₂₇	1	2	yes	—
206	C₂²	—	2	4	no	—
207	C₂²	—	2	4	no	—
208	C₂²	—	2	4	no	—
209	C₂²	—	2	4	no	—
210	C₂²	—	2	4	no	—
211	C₂²	—	2	4	no	—
212	C₂²	C₂×C₄⋊C₄×C₂₂₇	1	8	yes	complemented
213	C₂²	C₂³×C₄×C₂₂₇	1	1	yes	characteristic
214	C₂²	C₂²×D₄×C₂₂₇	1	1	yes	characteristic
215	C₂²	C₂²×D₄×C₂₂₇	1	1	yes	characteristic
216	C₂²	C₂²×Q₈×C₂₂₇	1	1	yes	characteristic
217	C₂²	C₂×#41×C₂₂₇	1	1	yes	characteristic
218	C₂²	H(5,𝔽₂)×C₂₂₇	1	1	yes	characteristic
219	C₂²	2₋¹⁺⁴×C₂₂₇	1	1	yes	characteristic
220	C₄	—	2	4	no	—
221	C₄	—	2	2	no	—
222	C₄	—	2	4	no	—
223	C₄	—	4	4	no	—
224	C₄	—	4	2	no	—
225	C₄	C₂×C₄⋊C₄×C₂₂₇	1	4	yes	complemented
226	C₄	C₄×D₄×C₂₂₇	1	4	yes	complemented
227	C₂	—	2	8	no	—
228	C₂	C₂²×C₄⋊C₄×C₂₂₇	1	1	yes	characteristic
229	C₂	C₂×C₄×D₄×C₂₂₇	1	1	yes	characteristic
230	C₂	#520×C₂₂₇	1	1	yes	characteristic
231	C₂	D₄²×C₂₂₇	1	1	yes	characteristic
232	C₂	#547×C₂₂₇	1	1	yes	characteristic
233	C₂	D₄×Q₈×C₂₂₇	1	1	yes	characteristic
234	C₂	#554×C₂₂₇	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₂₂₇	C₂⁴
commutator	C₂²	C₂³×C₄×C₂₂₇
Frattini, cosocle	C₂³	C₂⁴×C₂₂₇
prime radical	C₂²×D₄×C₂₂₇	C₂²
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³×C₂₂₇ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₂²×C₄⋊C₄×C₂₂₇	C₂	semidirect
4. #389×C₂₂₇	C₂	semidirect
5. #385×C₂₂₇	C₂	semidirect
6. #384×C₂₂₇	C₂	semidirect
7. #380×C₂₂₇	C₂	semidirect
8. C₄×C₄⋊C₄×C₂₂₇	C₂	semidirect
13. C₄×D₄×C₂₂₇	C₄	semidirect
37. C₂²×C₄×C₂₂₇	C₂×C₄	semidirect
50. C₄⋊C₄×C₂₂₇	C₂×C₄	semidirect
51. C₄⋊C₄×C₂₂₇	D₄	direct (N × H)
53. C₂²⋊C₄×C₂₂₇	C₂×C₄	semidirect
55. C₄²×C₂₂₇	C₂×C₄	semidirect
64. D₄×C₂₂₇	C₄⋊C₄	direct (N × H)
94. C₂²×C₂₂₇	C₂×C₄⋊C₄	semidirect
107. C₄×C₂₂₇	C₂×C₄⋊C₄	semidirect
108. C₄×C₂₂₇	C₄×D₄	semidirect
117. C₂₂₇	D₄×C₄⋊C₄	direct (N × H)
118. D₄×C₄⋊C₄	C₂₂₇	direct (N × H)
121. C₂²×C₄⋊C₄	C₂×C₂₂₇	semidirect
122. #389	C₂×C₂₂₇	semidirect
123. #385	C₂×C₂₂₇	semidirect
124. #384	C₂×C₂₂₇	semidirect
125. #380	C₂×C₂₂₇	semidirect
126. C₄×C₄⋊C₄	C₂×C₂₂₇	semidirect
131. C₄×D₄	C₄×C₂₂₇	semidirect
155. C₂²×C₄	C₂×C₄×C₂₂₇	semidirect
168. C₄⋊C₄	C₂×C₄×C₂₂₇	semidirect
169. C₄⋊C₄	D₄×C₂₂₇	direct (N × H)
171. C₂²⋊C₄	C₂×C₄×C₂₂₇	semidirect
173. C₄²	C₂×C₄×C₂₂₇	semidirect
182. D₄	C₄⋊C₄×C₂₂₇	direct (N × H)
212. C₂²	C₂×C₄⋊C₄×C₂₂₇	semidirect
225. C₄	C₂×C₄⋊C₄×C₂₂₇	semidirect
226. C₄	C₄×D₄×C₂₂₇	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄×D₄×C₂₂₇	2
C₂²×C₄⋊C₄×C₂₂₇	2
#389×C₂₂₇	2
#385×C₂₂₇	2
#384×C₂₂₇	2
#380×C₂₂₇	2
C₄×C₄⋊C₄×C₂₂₇	2
D₄×C₄⋊C₄	227

Subgroups — D4×C4⋊C4×C227 #44396304

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — D₄×C₄⋊C₄×C₂₂₇ #44396304