Subgroups — C₄⋊C₄×D₃₈₈ #27144639

Order 12416.

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×D₃₈₈	C₂	1	2	yes	maximal
2	C₂×C₄×D₃₈₈	C₂	1	1	yes	maximal, characteristic
3	C₂×C₄⋊C₄×D₉₇	C₂	1	2	yes	maximal, complemented
4	#20260633	C₂	1	2	yes	maximal, complemented
5	#20260609	C₂	1	4	yes	maximal, complemented
6	#20260605	C₂	1	1	yes	maximal, characteristic, complemented
7	#20260592	C₂	1	2	yes	maximal, complemented
8	C₄×C₄⋊C₄×C₉₇	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂²×D₃₈₈	C₂²	1	1	yes	characteristic
10	C₂²×C₄×D₉₇	C₂²	1	4	yes	—
11	C₂²×C₄×D₉₇	C₂²	1	2	yes	—
12	C₄×D₃₈₈	—	2	8	no	—
13	C₄×D₃₈₈	C₄	1	8	yes	complemented
14	C₂×C₄⋊Dic₉₇	C₂²	1	2	yes	—
15	C₂×C₄⋊Dic₉₇	C₂²	1	2	yes	—
16	C₂×C₄⋊Dic₉₇	C₂²	1	2	yes	—
17	C₂×C₄⋊Dic₉₇	C₂²	1	1	yes	characteristic
18	C₂×#13095490	C₂²	1	4	yes	—
19	C₄⋊C₄×D₉₇	—	2	8	no	—
20	C₂×#13095488	C₂²	1	4	yes	—
21	C₂×#13095488	C₂²	1	2	yes	—
22	#15690717	C₂²	1	4	yes	—
23	C₂×C₄⋊C₄×C₉₇	C₂²	1	1	yes	characteristic
24	C₂×C₄⋊C₄×C₉₇	C₂²	1	1	yes	characteristic
25	C₂×C₄²×C₉₇	C₂²	1	2	yes	—
26	C₂×C₄²×C₉₇	C₂²	1	1	yes	characteristic
27	#95×C₉₇	C₂²	1	2	yes	—
28	C₂³×D₉₇	C₂³	1	2	yes	—
29	C₂×D₃₈₈	C₂×C₄	1	4	yes	—
30	C₂×D₃₈₈	D₄	1	4	yes	—
31	C₂×D₃₈₈	Q₈	1	4	yes	—
32	C₂²×Dic₉₇	C₂³	1	4	yes	—
33	C₂²×Dic₉₇	C₂³	1	2	yes	—
34	C₂×C₄×D₉₇	—	2	4	no	—
35	C₂×C₄×D₉₇	—	2	4	no	—
36	C₂×C₄×D₉₇	—	2	8	no	—
37	C₂×C₄×D₉₇	—	2	8	no	—
38	C₂×C₄×D₉₇	—	4	4	no	—
39	C₂×C₄×D₉₇	C₂×C₄	1	8	yes	complemented
40	C₄⋊Dic₉₇	—	2	4	no	—
41	C₄⋊Dic₉₇	—	2	4	no	—
42	C₄⋊Dic₉₇	—	4	2	no	—
43	C₄⋊Dic₉₇	C₂×C₄	1	4	yes	complemented
44	#13095490	—	2	8	no	—
45	#13095488	—	2	8	no	—
46	#13095488	C₂×C₄	1	8	yes	complemented
47	C₂²×C₄×C₉₇	C₂³	1	2	yes	—
48	C₂²×C₄×C₉₇	C₂³	1	1	yes	characteristic
49	C₂²×C₄×C₉₇	C₂³	1	2	yes	—
50	C₂²×C₄×C₉₇	C₂³	1	1	yes	characteristic
51	C₂²×C₄×C₉₇	C₂³	1	1	yes	characteristic
52	C₄⋊C₄×C₉₇	—	2	2	no	—
53	C₄⋊C₄×C₉₇	D₄	1	4	yes	complemented
54	C₄²×C₉₇	—	2	4	no	—
55	C₄²×C₉₇	C₂×C₄	1	4	yes	complemented
56	C₂³×C₉₇	C₂⁴	1	1	yes	characteristic
57	C₂²×D₉₇	—	2	2	no	—
58	C₂²×D₉₇	—	2	2	no	—
59	C₂²×D₉₇	—	2	2	no	—
60	C₂²×D₉₇	—	2	2	no	—
61	C₂²×D₉₇	C₂²×C₄	1	4	yes	—
62	C₂²×D₉₇	C₂×D₄	1	4	yes	—
63	C₂²×D₉₇	C₂×Q₈	1	4	yes	—
64	C₂×C₄×C₉₇	—	2	2	no	—
65	C₂×C₄×C₉₇	—	2	1	no	—
66	C₂×C₄×C₉₇	—	2	2	no	—
67	C₂×C₄×C₉₇	—	2	1	no	—
68	C₂×C₄×C₉₇	—	2	2	no	—
69	C₂×C₄×C₉₇	—	2	2	no	—
70	C₂×C₄×C₉₇	—	2	2	no	—
71	C₂×C₄×C₉₇	C₂²×C₄	1	2	yes	—
72	C₂×C₄×C₉₇	C₂²×C₄	1	2	yes	—
73	C₂×C₄×C₉₇	C₂²×C₄	1	2	yes	—
74	C₂×C₄×C₉₇	C₂×D₄	1	2	yes	—
75	C₂×C₄×C₉₇	C₂×D₄	1	4	yes	—
76	C₂×C₄×C₉₇	C₂×D₄	1	2	yes	—
77	C₂×C₄×C₉₇	C₂×Q₈	1	2	yes	—
78	C₂×C₄×C₉₇	#41	1	2	yes	—
79	C₂×Dic₉₇	—	2	2	no	—
80	C₂×Dic₉₇	—	2	2	no	—
81	C₂×Dic₉₇	—	2	4	no	—
82	C₂×Dic₉₇	—	2	4	no	—
83	C₂×Dic₉₇	—	2	4	no	—
84	C₂×Dic₉₇	C₂²×C₄	1	4	yes	—
85	D₃₈₈	C₄⋊C₄	1	16	yes	direct product factor
86	C₄×D₉₇	—	2	16	no	—
87	C₄×D₉₇	—	4	16	no	—
88	C₂²×C₉₇	C₂³×C₄	1	1	yes	characteristic
89	C₂²×C₉₇	C₂²×D₄	1	1	yes	characteristic
90	C₂²×C₉₇	C₂²×D₄	1	1	yes	characteristic
91	C₂²×C₉₇	C₂²×Q₈	1	1	yes	characteristic
92	C₂²×C₉₇	C₂×#41	1	1	yes	characteristic
93	C₂²×C₉₇	H(5,𝔽₂)	1	1	yes	characteristic
94	C₂²×C₉₇	2₋¹⁺⁴	1	1	yes	characteristic
95	C₂×D₉₇	—	2	4	no	—
96	C₂×D₉₇	—	2	4	no	—
97	C₂×D₉₇	—	2	4	no	—
98	C₂×D₉₇	—	2	4	no	—
99	C₂×D₉₇	—	2	4	no	—
100	C₂×D₉₇	—	2	4	no	—
101	C₂×D₉₇	C₂×C₄⋊C₄	1	8	yes	complemented
102	C₄×C₉₇	—	2	2	no	—
103	C₄×C₉₇	—	2	4	no	—
104	C₄×C₉₇	—	4	2	no	—
105	C₄×C₉₇	C₂×C₄⋊C₄	1	4	yes	complemented
106	C₄×C₉₇	C₄×D₄	1	4	yes	complemented
107	Dic₉₇	—	2	4	no	—
108	Dic₉₇	—	4	4	no	—
109	C₂×C₉₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
110	C₂×C₉₇	C₂×C₄×D₄	1	1	yes	characteristic
111	C₂×C₉₇	#520	1	1	yes	characteristic
112	C₂×C₉₇	D₄²	1	1	yes	characteristic
113	C₂×C₉₇	#547	1	1	yes	characteristic
114	C₂×C₉₇	D₄×Q₈	1	1	yes	characteristic
115	C₂×C₉₇	#554	1	1	yes	characteristic
116	D₉₇	—	2	8	no	—
117	D₄×C₄⋊C₄	—	97	1	no	maximal, Sylow
118	C₉₇	D₄×C₄⋊C₄	1	1	yes	characteristic, complemented, Sylow
119	C₂×C₄×D₄	—	97	1	no	—
120	C₂×C₄×D₄	—	97	2	no	—
121	C₂²×C₄⋊C₄	—	97	2	no	—
122	#389	—	97	2	no	—
123	#385	—	97	4	no	—
124	#384	—	97	1	no	—
125	#380	—	97	2	no	—
126	C₄×C₄⋊C₄	D₉₇	1	1	yes	characteristic, complemented
127	C₂²×D₄	—	97	1	no	—
128	C₂³×C₄	—	97	2	no	—
129	C₂³×C₄	—	97	4	no	—
130	C₄×D₄	—	97	8	no	—
131	C₄×D₄	—	194	8	no	—
132	C₂×C₄⋊C₄	—	97	1	no	—
133	C₂×C₄⋊C₄	—	97	2	no	—
134	C₂×C₄⋊C₄	—	97	4	no	—
135	C₂×C₄⋊C₄	—	97	2	no	—
136	C₂×C₄⋊C₄	—	97	2	no	—
137	C₂×C₄⋊C₄	—	194	8	no	—
138	C₂×C₄⋊C₄	C₂×D₉₇	1	1	yes	characteristic
139	C₂×C₄⋊C₄	C₂×D₉₇	1	1	yes	characteristic
140	C₂×C₂²⋊C₄	—	97	2	no	—
141	C₂×C₂²⋊C₄	—	97	4	no	—
142	C₂×C₄²	C₂×D₉₇	1	2	yes	—
143	C₂×C₄²	C₂×D₉₇	1	1	yes	characteristic
144	#95	—	97	4	no	—
145	#95	C₂×D₉₇	1	2	yes	—
146	C₂⁴	—	97	2	no	—
147	C₂×D₄	—	97	4	no	—
148	C₂×D₄	—	97	4	no	—
149	C₂×D₄	—	97	4	no	—
150	C₂²×C₄	—	97	2	no	—
151	C₂²×C₄	—	97	8	no	—
152	C₂²×C₄	—	97	4	no	—
153	C₂²×C₄	—	388	4	no	—
154	C₂²×C₄	—	194	8	no	—
155	C₂²×C₄	—	194	8	no	—
156	C₂²×C₄	—	194	4	no	—
157	C₂²×C₄	—	194	4	no	—
158	C₂²×C₄	C₂²×D₉₇	1	2	yes	—
159	C₂²×C₄	C₂²×D₉₇	1	1	yes	characteristic
160	C₂²×C₄	C₂²×D₉₇	1	2	yes	—
161	C₂²×C₄	C₂²×D₉₇	1	1	yes	characteristic
162	C₂²×C₄	C₂²×D₉₇	1	1	yes	characteristic
163	C₄⋊C₄	—	2	2	no	—
164	C₄⋊C₄	—	97	4	no	—
165	C₄⋊C₄	—	388	2	no	—
166	C₄⋊C₄	—	194	4	no	—
167	C₄⋊C₄	—	194	8	no	—
168	C₄⋊C₄	—	194	4	no	—
169	C₄⋊C₄	D₃₈₈	1	4	yes	direct product factor
170	C₂²⋊C₄	—	97	8	no	—
171	C₂²⋊C₄	—	194	8	no	—
172	C₄²	—	2	4	no	—
173	C₄²	C₄×D₉₇	1	4	yes	complemented
174	C₂³	—	97	4	no	—
175	C₂³	—	97	4	no	—
176	C₂³	—	97	4	no	—
177	C₂³	—	194	2	no	—
178	C₂³	—	194	2	no	—
179	C₂³	—	194	2	no	—
180	C₂³	—	194	2	no	—
181	C₂³	C₂³×D₉₇	1	1	yes	characteristic
182	D₄	—	97	16	no	—
183	C₂×C₄	—	2	2	no	—
184	C₂×C₄	—	2	1	no	—
185	C₂×C₄	—	2	2	no	—
186	C₂×C₄	—	2	1	no	—
187	C₂×C₄	—	2	2	no	—
188	C₂×C₄	—	2	2	no	—
189	C₂×C₄	—	2	2	no	—
190	C₂×C₄	—	97	4	no	—
191	C₂×C₄	—	388	16	no	—
192	C₂×C₄	—	194	4	no	—
193	C₂×C₄	—	194	4	no	—
194	C₂×C₄	—	194	4	no	—
195	C₂×C₄	—	194	2	no	—
196	C₂×C₄	—	194	2	no	—
197	C₂×C₄	—	194	16	no	—
198	C₂×C₄	C₂×C₄×D₉₇	1	2	yes	—
199	C₂×C₄	C₂×C₄×D₉₇	1	2	yes	—
200	C₂×C₄	C₂×C₄×D₉₇	1	2	yes	—
201	C₂×C₄	D₄×D₉₇	1	2	yes	—
202	C₂×C₄	C₂×D₃₈₈	1	4	yes	—
203	C₂×C₄	C₂×D₃₈₈	1	2	yes	—
204	C₂×C₄	Q₈×D₉₇	1	2	yes	—
205	C₂×C₄	#13095511	1	2	yes	—
206	C₂²	—	97	8	no	—
207	C₂²	—	194	4	no	—
208	C₂²	—	194	4	no	—
209	C₂²	—	194	4	no	—
210	C₂²	—	194	4	no	—
211	C₂²	—	194	4	no	—
212	C₂²	—	194	4	no	—
213	C₂²	C₂²×C₄×D₉₇	1	1	yes	characteristic
214	C₂²	C₂×D₄×D₉₇	1	1	yes	characteristic
215	C₂²	C₂²×D₃₈₈	1	1	yes	characteristic
216	C₂²	C₂×Q₈×D₉₇	1	1	yes	characteristic
217	C₂²	C₂×#13095511	1	1	yes	characteristic
218	C₂²	#15690856	1	1	yes	characteristic
219	C₂²	#15690857	1	1	yes	characteristic
220	C₄	—	2	2	no	—
221	C₄	—	2	4	no	—
222	C₄	—	4	2	no	—
223	C₄	—	388	4	no	—
224	C₄	—	194	4	no	—
225	C₄	C₄⋊C₄×D₉₇	1	4	yes	complemented
226	C₄	C₄×D₃₈₈	1	4	yes	complemented
227	C₂	—	194	8	no	—
228	C₂	C₂×C₄⋊C₄×D₉₇	1	1	yes	characteristic
229	C₂	C₂×C₄×D₃₈₈	1	1	yes	characteristic
230	C₂	#20261215	1	1	yes	characteristic
231	C₂	D₄×D₃₈₈	1	1	yes	characteristic
232	C₂	#20261390	1	1	yes	characteristic
233	C₂	Q₈×D₃₈₈	1	1	yes	characteristic
234	C₂	#20261445	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₉₇
commutator	C₂²×C₉₇	C₂³×C₄
Fitting	C₄×C₄⋊C₄×C₉₇	C₂
socle, cosocle	C₂³×C₉₇	C₂⁴
prime radical	C₂²×D₃₈₈	C₂²
hypercenter	C₄×C₄⋊C₄	D₉₇
hypocenter	C₉₇	D₄×C₄⋊C₄
LCS	G ▷ C₂²×C₉₇ ▷ C₉₇
UCS	C₁ ◁ C₂³ ◁ C₄×C₄⋊C₄

Complements

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

N	H ≅ G/N	type
3. C₂×C₄⋊C₄×D₉₇	C₂	semidirect
4. #20260633	C₂	semidirect
5. #20260609	C₂	semidirect
6. #20260605	C₂	semidirect
7. #20260592	C₂	semidirect
8. C₄×C₄⋊C₄×C₉₇	C₂	semidirect
13. C₄×D₃₈₈	C₄	semidirect
39. C₂×C₄×D₉₇	C₂×C₄	semidirect
43. C₄⋊Dic₉₇	C₂×C₄	semidirect
46. #13095488	C₂×C₄	semidirect
53. C₄⋊C₄×C₉₇	D₄	semidirect
55. C₄²×C₉₇	C₂×C₄	semidirect
85. D₃₈₈	C₄⋊C₄	direct (N × H)
101. C₂×D₉₇	C₂×C₄⋊C₄	semidirect
105. C₄×C₉₇	C₂×C₄⋊C₄	semidirect
106. C₄×C₉₇	C₄×D₄	semidirect
118. C₉₇	D₄×C₄⋊C₄	semidirect
126. C₄×C₄⋊C₄	D₉₇	semidirect
169. C₄⋊C₄	D₃₈₈	direct (N × H)
173. C₄²	C₄×D₉₇	semidirect
225. C₄	C₄⋊C₄×D₉₇	semidirect
226. C₄	C₄×D₃₈₈	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄×D₃₈₈	2
C₂×C₄⋊C₄×D₉₇	2
#20260633	2
#20260609	2
#20260605	2
#20260592	2
C₄×C₄⋊C₄×C₉₇	2
D₄×C₄⋊C₄	97

Subgroups — C4⋊C4×D388 #27144639

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄⋊C₄×D₃₈₈ #27144639