Subgroups — C₄×#15690809 #27144306

Order 12416.

588 conjugacy classes of subgroups.

The subgroup lattice has 260 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#15690809	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄×Dic₁₉₄	C₂	1	2	yes	maximal
3	#20260639	C₂	1	2	yes	maximal
4	#20260634	C₂	1	1	yes	maximal, characteristic
5	#20260632	C₂	1	2	yes	maximal
6	#20260631	C₂	1	1	yes	maximal, characteristic
7	#20260604	C₂	1	1	yes	maximal, characteristic
8	C₄×#13095490	C₂	1	2	yes	maximal
9	C₄×C₄⋊Dic₉₇	C₂	1	1	yes	maximal, characteristic
10	C₄²×Dic₉₇	C₂	1	1	yes	maximal, characteristic
11	C₄×C₄⋊C₄×C₉₇	C₂	1	1	yes	maximal, characteristic
12	C₂²×Dic₁₉₄	C₂²	1	2	yes	—
13	#15690809	C₄	1	8	yes	direct product factor
14	C₄×Dic₁₉₄	—	2	8	no	—
15	C₂×C₄⋊Dic₉₇	C₂²	1	2	yes	—
16	C₂×C₄⋊Dic₉₇	C₂²	1	1	yes	characteristic
17	C₂×C₄⋊Dic₉₇	C₂²	1	1	yes	characteristic
18	C₂×#13095490	C₂²	1	2	yes	—
19	C₂×#13095490	C₂²	1	2	yes	—
20	C₂×#13095490	C₂²	1	2	yes	—
21	C₂×#13095490	C₂²	1	2	yes	—
22	C₂×C₄×Dic₉₇	C₂²	1	1	yes	characteristic
23	C₂×C₄×Dic₉₇	C₂²	1	1	yes	characteristic
24	C₂×C₄×Dic₉₇	C₂²	1	1	yes	characteristic
25	C₂×C₄×Dic₉₇	C₂²	1	2	yes	—
26	C₂×C₄×Dic₉₇	C₂²	1	1	yes	characteristic
27	C₂×C₄×Dic₉₇	C₂²	1	1	yes	characteristic
28	C₂×C₄×Dic₉₇	C₂²	1	1	yes	characteristic
29	#15690717	C₂²	1	2	yes	—
30	#15690717	C₂²	1	2	yes	—
31	#15690717	C₂²	1	2	yes	—
32	C₂×C₄⋊C₄×C₉₇	C₂²	1	1	yes	characteristic
33	C₂×C₄⋊C₄×C₉₇	C₂²	1	1	yes	characteristic
34	C₂×C₄²×C₉₇	C₂²	1	2	yes	—
35	C₂×C₄²×C₉₇	C₂²	1	1	yes	characteristic
36	#95×C₉₇	C₂²	1	2	yes	—
37	C₂×Dic₁₉₄	—	2	4	no	—
38	C₂×Dic₁₉₄	—	2	4	no	—
39	C₂×Dic₁₉₄	C₂×C₄	1	8	yes	—
40	C₂²×Dic₉₇	C₂³	1	2	yes	—
41	C₂²×Dic₉₇	C₂³	1	2	yes	—
42	C₂²×Dic₉₇	C₂³	1	1	yes	characteristic
43	C₂²×Dic₉₇	C₂³	1	1	yes	characteristic
44	C₂²×Dic₉₇	C₂³	1	1	yes	characteristic
45	C₂²×Dic₉₇	C₂³	1	1	yes	characteristic
46	C₄⋊Dic₉₇	—	2	2	no	—
47	C₄⋊Dic₉₇	—	2	4	no	—
48	C₄⋊Dic₉₇	C₂×C₄	1	4	yes	—
49	#13095490	—	2	4	no	—
50	#13095490	—	2	4	no	—
51	#13095490	—	2	4	no	—
52	#13095490	C₂×C₄	1	8	yes	—
53	C₄×Dic₉₇	—	2	4	no	—
54	C₄×Dic₉₇	—	2	2	no	—
55	C₄×Dic₉₇	—	2	2	no	—
56	C₄×Dic₉₇	—	2	2	no	—
57	C₄×Dic₉₇	C₂×C₄	1	4	yes	—
58	C₄×Dic₉₇	D₄	1	4	yes	—
59	C₄×Dic₉₇	Q₈	1	4	yes	complemented
60	C₂²×C₄×C₉₇	C₂³	1	2	yes	—
61	C₂²×C₄×C₉₇	C₂³	1	2	yes	—
62	C₂²×C₄×C₉₇	C₂³	1	1	yes	characteristic
63	C₂²×C₄×C₉₇	C₂³	1	1	yes	characteristic
64	C₂²×C₄×C₉₇	C₂³	1	1	yes	characteristic
65	C₄⋊C₄×C₉₇	—	2	2	no	—
66	C₄⋊C₄×C₉₇	C₂×C₄	1	4	yes	—
67	C₄²×C₉₇	—	2	4	no	—
68	C₄²×C₉₇	Q₈	1	4	yes	complemented
69	C₂³×C₉₇	C₂⁴	1	1	yes	characteristic
70	C₂×C₄×C₉₇	—	2	2	no	—
71	C₂×C₄×C₉₇	—	2	1	no	—
72	C₂×C₄×C₉₇	—	2	1	no	—
73	C₂×C₄×C₉₇	—	2	2	no	—
74	C₂×C₄×C₉₇	—	2	2	no	—
75	C₂×C₄×C₉₇	—	2	2	no	—
76	C₂×C₄×C₉₇	—	2	2	no	—
77	C₂×C₄×C₉₇	C₂²×C₄	1	4	yes	—
78	C₂×C₄×C₉₇	C₂²×C₄	1	2	yes	—
79	C₂×C₄×C₉₇	C₂×D₄	1	2	yes	—
80	C₂×C₄×C₉₇	C₂×Q₈	1	2	yes	—
81	C₂×C₄×C₉₇	C₂×Q₈	1	2	yes	—
82	C₂×C₄×C₉₇	C₂×Q₈	1	2	yes	—
83	C₂×C₄×C₉₇	C₂×Q₈	1	2	yes	—
84	C₂×C₄×C₉₇	#41	1	2	yes	—
85	C₂×Dic₉₇	—	2	2	no	—
86	C₂×Dic₉₇	—	2	1	no	—
87	C₂×Dic₉₇	—	2	1	no	—
88	C₂×Dic₉₇	—	2	1	no	—
89	C₂×Dic₉₇	—	2	1	no	—
90	C₂×Dic₉₇	—	2	2	no	—
91	C₂×Dic₉₇	—	2	2	no	—
92	C₂×Dic₉₇	—	2	2	no	—
93	C₂×Dic₉₇	—	2	2	no	—
94	C₂×Dic₉₇	C₂²×C₄	1	4	yes	—
95	C₂×Dic₉₇	C₂²×C₄	1	2	yes	—
96	C₂×Dic₉₇	C₂²×C₄	1	2	yes	—
97	C₂×Dic₉₇	C₂×D₄	1	2	yes	—
98	C₂×Dic₉₇	C₂×D₄	1	2	yes	—
99	C₂×Dic₉₇	C₂×Q₈	1	2	yes	—
100	C₂×Dic₉₇	C₂×Q₈	1	2	yes	—
101	C₂×Dic₉₇	#41	1	2	yes	—
102	C₂×Dic₉₇	#41	1	2	yes	—
103	Dic₁₉₄	—	2	16	no	—
104	C₂²×C₉₇	C₂³×C₄	1	1	yes	characteristic
105	C₂²×C₉₇	C₂²×D₄	1	1	yes	characteristic
106	C₂²×C₉₇	C₂²×Q₈	1	1	yes	characteristic
107	C₂²×C₉₇	C₂²×Q₈	1	1	yes	characteristic
108	C₂²×C₉₇	C₂×#41	1	1	yes	characteristic
109	C₂²×C₉₇	C₂×#41	1	1	yes	characteristic
110	C₂²×C₉₇	C₂×#41	1	1	yes	characteristic
111	C₄×C₉₇	—	2	4	no	—
112	C₄×C₉₇	—	2	2	no	—
113	C₄×C₉₇	—	4	2	no	—
114	C₄×C₉₇	C₄×Q₈	1	4	yes	complemented
115	C₄×C₉₇	C₄⋊Q₈	1	4	yes	complemented
116	Dic₉₇	—	2	4	no	—
117	Dic₉₇	—	2	2	no	—
118	Dic₉₇	—	2	2	no	—
119	Dic₉₇	—	4	2	no	—
120	Dic₉₇	C₄×D₄	1	4	yes	—
121	Dic₉₇	C₄×Q₈	1	4	yes	complemented
122	C₂×C₉₇	C₂×C₄×D₄	1	1	yes	characteristic
123	C₂×C₉₇	C₂×C₄×Q₈	1	1	yes	characteristic
124	C₂×C₉₇	C₂×C₄×Q₈	1	1	yes	characteristic
125	C₂×C₉₇	C₂×C₄⋊Q₈	1	1	yes	characteristic
126	C₂×C₉₇	#532	1	1	yes	characteristic
127	C₂×C₉₇	#533	1	1	yes	characteristic
128	C₂×C₉₇	#533	1	1	yes	characteristic
129	C₄×C₄⋊Q₈	—	97	1	no	maximal, Sylow
130	C₉₇	C₄×C₄⋊Q₈	1	1	yes	characteristic, complemented, Sylow
131	C₂×C₄⋊Q₈	—	97	1	no	—
132	C₂×C₄×Q₈	—	97	2	no	—
133	#391	—	97	2	no	—
134	#389	—	97	1	no	—
135	#389	—	97	1	no	—
136	#389	—	97	2	no	—
137	#384	—	97	1	no	—
138	C₄×C₄⋊C₄	—	97	1	no	—
139	C₄×C₄⋊C₄	—	97	2	no	—
140	C₄×C₄⋊C₄	D₉₇	1	1	yes	characteristic
141	C₄³	—	97	1	no	—
142	C₂²×Q₈	—	97	2	no	—
143	C₄⋊Q₈	—	97	8	no	—
144	C₄×Q₈	—	194	8	no	—
145	C₂×C₄⋊C₄	—	97	2	no	—
146	C₂×C₄⋊C₄	—	97	1	no	—
147	C₂×C₄⋊C₄	—	97	2	no	—
148	C₂×C₄⋊C₄	—	97	2	no	—
149	C₂×C₄⋊C₄	—	97	2	no	—
150	C₂×C₄⋊C₄	—	97	2	no	—
151	C₂×C₄⋊C₄	—	97	1	no	—
152	C₂×C₄⋊C₄	C₂×D₉₇	1	1	yes	characteristic
153	C₂×C₄⋊C₄	C₂×D₉₇	1	1	yes	characteristic
154	C₂×C₄²	—	97	1	no	—
155	C₂×C₄²	—	97	1	no	—
156	C₂×C₄²	—	97	1	no	—
157	C₂×C₄²	—	97	1	no	—
158	C₂×C₄²	—	97	1	no	—
159	C₂×C₄²	—	97	1	no	—
160	C₂×C₄²	—	97	2	no	—
161	C₂×C₄²	C₂×D₉₇	1	2	yes	—
162	C₂×C₄²	C₂×D₉₇	1	1	yes	characteristic
163	#95	—	97	2	no	—
164	#95	—	97	2	no	—
165	#95	—	97	2	no	—
166	#95	C₂×D₉₇	1	2	yes	—
167	C₂×Q₈	—	97	8	no	—
168	C₂×Q₈	—	194	4	no	—
169	C₂×Q₈	—	194	4	no	—
170	C₂²×C₄	—	97	1	no	—
171	C₂²×C₄	—	97	1	no	—
172	C₂²×C₄	—	97	1	no	—
173	C₂²×C₄	—	97	2	no	—
174	C₂²×C₄	—	97	1	no	—
175	C₂²×C₄	—	97	2	no	—
176	C₂²×C₄	C₂²×D₉₇	1	2	yes	—
177	C₂²×C₄	C₂²×D₉₇	1	2	yes	—
178	C₂²×C₄	C₂²×D₉₇	1	1	yes	characteristic
179	C₂²×C₄	C₂²×D₉₇	1	1	yes	characteristic
180	C₂²×C₄	C₂²×D₉₇	1	1	yes	characteristic
181	C₄⋊C₄	—	2	2	no	—
182	C₄⋊C₄	—	97	8	no	—
183	C₄⋊C₄	—	97	4	no	—
184	C₄⋊C₄	—	194	4	no	—
185	C₄⋊C₄	—	194	4	no	—
186	C₄⋊C₄	—	194	4	no	—
187	C₄⋊C₄	—	194	4	no	—
188	C₄⋊C₄	—	194	2	no	—
189	C₄⋊C₄	C₄×D₉₇	1	4	yes	—
190	C₄²	—	2	4	no	—
191	C₄²	—	97	4	no	—
192	C₄²	—	97	4	no	—
193	C₄²	—	97	4	no	—
194	C₄²	—	194	2	no	—
195	C₄²	—	194	2	no	—
196	C₄²	—	194	4	no	—
197	C₄²	—	194	2	no	—
198	C₄²	Dic₁₉₄	1	4	yes	complemented
199	C₂³	C₂³×D₉₇	1	1	yes	characteristic
200	Q₈	—	194	16	no	—
201	C₂×C₄	—	2	2	no	—
202	C₂×C₄	—	2	1	no	—
203	C₂×C₄	—	2	1	no	—
204	C₂×C₄	—	2	2	no	—
205	C₂×C₄	—	2	2	no	—
206	C₂×C₄	—	2	2	no	—
207	C₂×C₄	—	2	2	no	—
208	C₂×C₄	—	97	2	no	—
209	C₂×C₄	—	97	2	no	—
210	C₂×C₄	—	97	2	no	—
211	C₂×C₄	—	97	2	no	—
212	C₂×C₄	—	97	2	no	—
213	C₂×C₄	—	97	2	no	—
214	C₂×C₄	—	97	4	no	—
215	C₂×C₄	—	97	2	no	—
216	C₂×C₄	—	97	2	no	—
217	C₂×C₄	—	194	2	no	—
218	C₂×C₄	—	194	2	no	—
219	C₂×C₄	—	194	2	no	—
220	C₂×C₄	—	194	2	no	—
221	C₂×C₄	—	194	1	no	—
222	C₂×C₄	—	194	1	no	—
223	C₂×C₄	—	194	2	no	—
224	C₂×C₄	—	194	1	no	—
225	C₂×C₄	—	194	1	no	—
226	C₂×C₄	C₂×C₄×D₉₇	1	4	yes	—
227	C₂×C₄	C₂×C₄×D₉₇	1	2	yes	—
228	C₂×C₄	D₄×D₉₇	1	2	yes	—
229	C₂×C₄	Q₈×D₉₇	1	2	yes	—
230	C₂×C₄	C₂×Dic₁₉₄	1	2	yes	—
231	C₂×C₄	C₂×Dic₁₉₄	1	2	yes	—
232	C₂×C₄	C₂×Dic₁₉₄	1	2	yes	—
233	C₂×C₄	#13095511	1	2	yes	—
234	C₂²	C₂²×C₄×D₉₇	1	1	yes	characteristic
235	C₂²	C₂×D₄×D₉₇	1	1	yes	characteristic
236	C₂²	C₂×Q₈×D₉₇	1	1	yes	characteristic
237	C₂²	C₂²×Dic₁₉₄	1	1	yes	characteristic
238	C₂²	#41×D₉₇	1	1	yes	characteristic
239	C₂²	#41×D₉₇	1	1	yes	characteristic
240	C₂²	C₂×#13095511	1	1	yes	characteristic
241	C₄	—	2	4	no	—
242	C₄	—	2	2	no	—
243	C₄	—	4	2	no	—
244	C₄	—	97	4	no	—
245	C₄	—	97	4	no	—
246	C₄	—	388	2	no	—
247	C₄	—	194	4	no	—
248	C₄	—	194	2	no	—
249	C₄	—	194	2	no	—
250	C₄	C₄×Dic₁₉₄	1	4	yes	complemented
251	C₄	#15690809	1	4	yes	direct product factor
252	C₂	C₄×D₄×D₉₇	1	1	yes	characteristic
253	C₂	C₄×Q₈×D₉₇	1	1	yes	characteristic
254	C₂	C₂×C₄×Dic₁₉₄	1	1	yes	characteristic
255	C₂	C₂×#15690809	1	1	yes	characteristic
256	C₂	#20261282	1	1	yes	characteristic
257	C₂	#20261288	1	1	yes	characteristic
258	C₂	#20261289	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₂²×D₉₇
commutator	C₂²×C₉₇	C₂³×C₄
Frattini	C₂³	C₂³×D₉₇
Fitting	C₄×C₄⋊C₄×C₉₇	C₂
socle, prime radical, cosocle	C₂³×C₉₇	C₂⁴
hypercenter	C₄×C₄⋊C₄	D₉₇
hypocenter	C₉₇	C₄×C₄⋊Q₈
LCS	G ▷ C₂²×C₉₇ ▷ C₉₇
UCS	C₁ ◁ C₂²×C₄ ◁ C₄×C₄⋊C₄

Complements

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

N	H ≅ G/N	type
13. #15690809	C₄	direct (N × H)
59. C₄×Dic₉₇	Q₈	semidirect
68. C₄²×C₉₇	Q₈	semidirect
114. C₄×C₉₇	C₄×Q₈	semidirect
115. C₄×C₉₇	C₄⋊Q₈	semidirect
121. Dic₉₇	C₄×Q₈	semidirect
130. C₉₇	C₄×C₄⋊Q₈	semidirect
198. C₄²	Dic₁₉₄	semidirect
250. C₄	C₄×Dic₁₉₄	semidirect
251. C₄	#15690809	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×#15690809	2
C₂×C₄×Dic₁₉₄	2
#20260639	2
#20260634	2
#20260632	2
#20260631	2
#20260604	2
C₄×#13095490	2
C₄×C₄⋊Dic₉₇	2
C₄²×Dic₉₇	2
C₄×C₄⋊C₄×C₉₇	2
C₄×C₄⋊Q₈	97

Subgroups — C4×#15690809 #27144306

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄×#15690809 #27144306