Subgroups — C₂×#464×C₉₇ #27138992

Order 12416.

500 conjugacy classes of subgroups.

The subgroup lattice has 232 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×Q₁₆×C₉₇	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂²×SD₁₆×C₉₇	C₂	1	1	yes	maximal, characteristic
3	C₂×#124×C₉₇	C₂	1	1	yes	maximal, characteristic
4	C₂×C₄×Q₈×C₉₇	C₂	1	1	yes	maximal, characteristic, complemented
5	#464×C₉₇	C₂	1	8	yes	maximal, direct product factor
6	C₂×C₄⋊C₈×C₉₇	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×#103×C₉₇	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#102×C₉₇	C₂	1	1	yes	maximal, characteristic
9	C₂²×Q₈×C₉₇	C₂²	1	1	yes	characteristic
10	C₂²×Q₈×C₉₇	C₂²	1	1	yes	characteristic
11	C₂²×D₄×C₉₇	C₂²	1	1	yes	characteristic
12	C₂×Q₁₆×C₉₇	—	2	4	no	—
13	C₂×Q₁₆×C₉₇	C₂²	1	4	yes	complemented
14	C₂×SD₁₆×C₉₇	—	2	4	no	—
15	C₂×SD₁₆×C₉₇	C₂²	1	4	yes	—
16	C₂²×C₈×C₉₇	C₂²	1	1	yes	characteristic
17	C₂²×C₈×C₉₇	C₂²	1	1	yes	characteristic
18	#124×C₉₇	—	2	2	no	—
19	#124×C₉₇	C₂²	1	4	yes	—
20	C₄×Q₈×C₉₇	—	2	2	no	—
21	C₄×Q₈×C₉₇	C₂²	1	4	yes	complemented
22	C₂×C₄⋊C₄×C₉₇	—	2	1	no	—
23	C₂×C₄⋊C₄×C₉₇	C₂²	1	1	yes	characteristic
24	C₂×C₂²⋊C₄×C₉₇	—	2	1	no	—
25	C₂×C₂²⋊C₄×C₉₇	—	2	1	no	—
26	C₂×C₄²×C₉₇	—	2	1	no	—
27	C₂×C₄²×C₉₇	C₂²	1	1	yes	characteristic
28	C₄⋊C₈×C₉₇	C₂²	1	4	yes	complemented
29	#103×C₉₇	C₂²	1	4	yes	complemented
30	#102×C₉₇	C₂²	1	4	yes	—
31	C₂⁴×C₉₇	—	2	1	no	—
32	C₂×Q₈×C₉₇	—	2	1	no	—
33	C₂×Q₈×C₉₇	—	2	1	no	—
34	C₂×Q₈×C₉₇	—	2	2	no	—
35	C₂×Q₈×C₉₇	—	2	2	no	—
36	C₂×Q₈×C₉₇	—	2	2	no	—
37	C₂×Q₈×C₉₇	D₄	1	4	yes	—
38	C₂×Q₈×C₉₇	C₂³	1	2	yes	—
39	C₂×Q₈×C₉₇	C₂³	1	2	yes	—
40	C₂×D₄×C₉₇	—	2	2	no	—
41	C₂×D₄×C₉₇	—	2	1	no	—
42	C₂×D₄×C₉₇	—	2	2	no	—
43	C₂×D₄×C₉₇	C₂³	1	2	yes	—
44	C₂²×C₄×C₉₇	—	2	1	no	—
45	C₂²×C₄×C₉₇	—	2	1	no	—
46	C₂²×C₄×C₉₇	—	2	1	no	—
47	C₂²×C₄×C₉₇	D₄	1	1	yes	characteristic
48	C₂²×C₄×C₉₇	D₄	1	1	yes	characteristic
49	C₂²×C₄×C₉₇	C₂³	1	1	yes	characteristic
50	Q₁₆×C₉₇	—	2	8	no	—
51	SD₁₆×C₉₇	—	2	8	no	—
52	C₂×C₈×C₉₇	—	2	2	no	—
53	C₂×C₈×C₉₇	—	2	2	no	—
54	C₂×C₈×C₉₇	C₂³	1	2	yes	—
55	C₂×C₈×C₉₇	C₂³	1	2	yes	—
56	C₄⋊C₄×C₉₇	—	2	4	no	—
57	C₄⋊C₄×C₉₇	—	2	1	no	—
58	C₄⋊C₄×C₉₇	C₂³	1	2	yes	—
59	C₂²⋊C₄×C₉₇	—	2	4	no	—
60	C₂²⋊C₄×C₉₇	—	2	4	no	—
61	C₄²×C₉₇	—	2	4	no	—
62	C₄²×C₉₇	D₄	1	2	yes	—
63	C₄²×C₉₇	C₂³	1	2	yes	—
64	C₂³×C₉₇	—	2	2	no	—
65	C₂³×C₉₇	—	4	2	no	—
66	C₂³×C₉₇	—	4	2	no	—
67	C₂³×C₉₇	—	4	2	no	—
68	C₂³×C₉₇	C₂×D₄	1	1	yes	characteristic
69	Q₈×C₉₇	—	2	4	no	—
70	Q₈×C₉₇	—	2	2	no	—
71	Q₈×C₉₇	—	2	2	no	—
72	Q₈×C₉₇	—	2	2	no	—
73	Q₈×C₉₇	—	4	2	no	—
74	Q₈×C₉₇	C₂×D₄	1	4	yes	—
75	D₄×C₉₇	—	2	2	no	—
76	D₄×C₉₇	—	2	2	no	—
77	D₄×C₉₇	—	4	2	no	—
78	C₂×C₄×C₉₇	—	2	4	no	—
79	C₂×C₄×C₉₇	—	2	2	no	—
80	C₂×C₄×C₉₇	—	2	2	no	—
81	C₂×C₄×C₉₇	—	2	2	no	—
82	C₂×C₄×C₉₇	—	2	2	no	—
83	C₂×C₄×C₉₇	—	2	2	no	—
84	C₂×C₄×C₉₇	—	4	2	no	—
85	C₂×C₄×C₉₇	—	4	2	no	—
86	C₂×C₄×C₉₇	C₂×D₄	1	2	yes	—
87	C₂×C₄×C₉₇	C₂×D₄	1	2	yes	—
88	C₂×C₄×C₉₇	C₂×D₄	1	1	yes	characteristic
89	C₂×C₄×C₉₇	C₂×D₄	1	2	yes	—
90	C₂×C₄×C₉₇	#41	1	2	yes	—
91	C₂×C₄×C₉₇	C₂⁴	1	1	yes	characteristic
92	C₈×C₉₇	—	2	2	no	—
93	C₈×C₉₇	—	2	2	no	—
94	C₂²×C₉₇	—	4	2	no	—
95	C₂²×C₉₇	—	4	2	no	—
96	C₂²×C₉₇	—	4	2	no	—
97	C₂²×C₉₇	—	8	4	no	—
98	C₂²×C₉₇	#121	1	2	yes	—
99	C₂²×C₉₇	#135	1	2	yes	—
100	C₂²×C₉₇	#137	1	2	yes	—
101	C₂²×C₉₇	C₂²×D₄	1	1	yes	characteristic
102	C₄×C₉₇	—	2	4	no	—
103	C₄×C₉₇	—	2	2	no	—
104	C₄×C₉₇	—	2	2	no	—
105	C₄×C₉₇	—	4	2	no	—
106	C₄×C₉₇	—	4	2	no	—
107	C₄×C₉₇	#121	1	2	yes	—
108	C₄×C₉₇	C₂²×D₄	1	1	yes	characteristic
109	C₄×C₉₇	C₂×#41	1	1	yes	characteristic
110	C₂×C₉₇	—	8	2	no	—
111	C₂×C₉₇	#464	1	4	yes	direct product factor
112	C₂×C₉₇	C₂×#121	1	1	yes	characteristic
113	C₂×C₉₇	C₂×#135	1	1	yes	characteristic
114	C₂×C₉₇	C₂×#137	1	1	yes	characteristic
115	C₂×#464	C₉₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
116	C₉₇	C₂×#464	1	1	yes	characteristic, direct product factor, Sylow
117	C₂²×Q₁₆	C₂×C₉₇	1	1	yes	characteristic, complemented
118	C₂²×SD₁₆	C₂×C₉₇	1	1	yes	characteristic
119	C₂×#124	C₂×C₉₇	1	1	yes	characteristic
120	C₂×C₄×Q₈	C₂×C₉₇	1	1	yes	characteristic, complemented
121	#464	C₂×C₉₇	1	8	yes	direct product factor
122	C₂×C₄⋊C₈	C₂×C₉₇	1	1	yes	characteristic, complemented
123	C₂×#103	C₂×C₉₇	1	1	yes	characteristic, complemented
124	C₂×#102	C₂×C₉₇	1	1	yes	characteristic
125	C₂²×Q₈	C₂²×C₉₇	1	1	yes	characteristic
126	C₂²×Q₈	C₂²×C₉₇	1	1	yes	characteristic
127	C₂²×D₄	C₂²×C₉₇	1	1	yes	characteristic
128	C₂×Q₁₆	—	2	4	no	—
129	C₂×Q₁₆	C₂²×C₉₇	1	4	yes	complemented
130	C₂×SD₁₆	—	2	4	no	—
131	C₂×SD₁₆	C₂²×C₉₇	1	4	yes	—
132	C₂²×C₈	C₂²×C₉₇	1	1	yes	characteristic
133	C₂²×C₈	C₂²×C₉₇	1	1	yes	characteristic
134	#124	—	2	2	no	—
135	#124	C₂²×C₉₇	1	4	yes	—
136	C₄×Q₈	—	2	2	no	—
137	C₄×Q₈	C₂²×C₉₇	1	4	yes	complemented
138	C₂×C₄⋊C₄	—	2	1	no	—
139	C₂×C₄⋊C₄	C₂²×C₉₇	1	1	yes	characteristic
140	C₂×C₂²⋊C₄	—	2	1	no	—
141	C₂×C₂²⋊C₄	—	2	1	no	—
142	C₂×C₄²	—	2	1	no	—
143	C₂×C₄²	C₂²×C₉₇	1	1	yes	characteristic
144	C₄⋊C₈	C₂²×C₉₇	1	4	yes	complemented
145	#103	C₂²×C₉₇	1	4	yes	complemented
146	#102	C₂²×C₉₇	1	4	yes	—
147	C₂⁴	—	2	1	no	—
148	C₂×Q₈	—	2	1	no	—
149	C₂×Q₈	—	2	1	no	—
150	C₂×Q₈	—	2	2	no	—
151	C₂×Q₈	—	2	2	no	—
152	C₂×Q₈	—	2	2	no	—
153	C₂×Q₈	D₄×C₉₇	1	4	yes	—
154	C₂×Q₈	C₂³×C₉₇	1	2	yes	—
155	C₂×Q₈	C₂³×C₉₇	1	2	yes	—
156	C₂×D₄	—	2	2	no	—
157	C₂×D₄	—	2	1	no	—
158	C₂×D₄	—	2	2	no	—
159	C₂×D₄	C₂³×C₉₇	1	2	yes	—
160	C₂²×C₄	—	2	1	no	—
161	C₂²×C₄	—	2	1	no	—
162	C₂²×C₄	—	2	1	no	—
163	C₂²×C₄	D₄×C₉₇	1	1	yes	characteristic
164	C₂²×C₄	D₄×C₉₇	1	1	yes	characteristic
165	C₂²×C₄	C₂³×C₉₇	1	1	yes	characteristic
166	Q₁₆	—	2	8	no	—
167	SD₁₆	—	2	8	no	—
168	C₂×C₈	—	2	2	no	—
169	C₂×C₈	—	2	2	no	—
170	C₂×C₈	C₂³×C₉₇	1	2	yes	—
171	C₂×C₈	C₂³×C₉₇	1	2	yes	—
172	C₄⋊C₄	—	2	4	no	—
173	C₄⋊C₄	—	2	1	no	—
174	C₄⋊C₄	C₂³×C₉₇	1	2	yes	—
175	C₂²⋊C₄	—	2	4	no	—
176	C₂²⋊C₄	—	2	4	no	—
177	C₄²	—	2	4	no	—
178	C₄²	D₄×C₉₇	1	2	yes	—
179	C₄²	C₂³×C₉₇	1	2	yes	—
180	C₂³	—	2	2	no	—
181	C₂³	—	4	2	no	—
182	C₂³	—	4	2	no	—
183	C₂³	—	4	2	no	—
184	C₂³	C₂×D₄×C₉₇	1	1	yes	characteristic
185	Q₈	—	2	4	no	—
186	Q₈	—	2	2	no	—
187	Q₈	—	2	2	no	—
188	Q₈	—	2	2	no	—
189	Q₈	—	4	2	no	—
190	Q₈	C₂×D₄×C₉₇	1	4	yes	—
191	D₄	—	2	2	no	—
192	D₄	—	2	2	no	—
193	D₄	—	4	2	no	—
194	C₂×C₄	—	2	4	no	—
195	C₂×C₄	—	2	2	no	—
196	C₂×C₄	—	2	2	no	—
197	C₂×C₄	—	2	2	no	—
198	C₂×C₄	—	2	2	no	—
199	C₂×C₄	—	2	2	no	—
200	C₂×C₄	—	4	2	no	—
201	C₂×C₄	—	4	2	no	—
202	C₂×C₄	C₂×D₄×C₉₇	1	2	yes	—
203	C₂×C₄	C₂×D₄×C₉₇	1	2	yes	—
204	C₂×C₄	C₂×D₄×C₉₇	1	1	yes	characteristic
205	C₂×C₄	C₂×D₄×C₉₇	1	2	yes	—
206	C₂×C₄	#41×C₉₇	1	2	yes	—
207	C₂×C₄	C₂⁴×C₉₇	1	1	yes	characteristic
208	C₈	—	2	2	no	—
209	C₈	—	2	2	no	—
210	C₂²	—	4	2	no	—
211	C₂²	—	4	2	no	—
212	C₂²	—	4	2	no	—
213	C₂²	—	8	4	no	—
214	C₂²	#121×C₉₇	1	2	yes	—
215	C₂²	#135×C₉₇	1	2	yes	—
216	C₂²	#137×C₉₇	1	2	yes	—
217	C₂²	C₂²×D₄×C₉₇	1	1	yes	characteristic
218	C₄	—	2	4	no	—
219	C₄	—	2	2	no	—
220	C₄	—	2	2	no	—
221	C₄	—	4	2	no	—
222	C₄	—	4	2	no	—
223	C₄	#121×C₉₇	1	2	yes	—
224	C₄	C₂²×D₄×C₉₇	1	1	yes	characteristic
225	C₄	C₂×#41×C₉₇	1	1	yes	characteristic
226	C₂	—	8	2	no	—
227	C₂	#464×C₉₇	1	4	yes	direct product factor
228	C₂	C₂×#121×C₉₇	1	1	yes	characteristic
229	C₂	C₂×#135×C₉₇	1	1	yes	characteristic
230	C₂	C₂×#137×C₉₇	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₉₇	C₂×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁴×C₉₇
prime radical	C₂²×D₄×C₉₇	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₉₇ ◁ C₂×C₄²×C₉₇ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂²×Q₁₆×C₉₇	C₂	semidirect
4. C₂×C₄×Q₈×C₉₇	C₂	semidirect
5. #464×C₉₇	C₂	direct (N × H)
6. C₂×C₄⋊C₈×C₉₇	C₂	semidirect
7. C₂×#103×C₉₇	C₂	semidirect
13. C₂×Q₁₆×C₉₇	C₂²	semidirect
21. C₄×Q₈×C₉₇	C₂²	semidirect
28. C₄⋊C₈×C₉₇	C₂²	semidirect
29. #103×C₉₇	C₂²	semidirect
111. C₂×C₉₇	#464	direct (N × H)
115. C₂×#464	C₉₇	direct (N × H)
116. C₉₇	C₂×#464	direct (N × H)
117. C₂²×Q₁₆	C₂×C₉₇	semidirect
120. C₂×C₄×Q₈	C₂×C₉₇	semidirect
121. #464	C₂×C₉₇	direct (N × H)
122. C₂×C₄⋊C₈	C₂×C₉₇	semidirect
123. C₂×#103	C₂×C₉₇	semidirect
129. C₂×Q₁₆	C₂²×C₉₇	semidirect
137. C₄×Q₈	C₂²×C₉₇	semidirect
144. C₄⋊C₈	C₂²×C₉₇	semidirect
145. #103	C₂²×C₉₇	semidirect
227. C₂	#464×C₉₇	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×Q₁₆×C₉₇	2
C₂²×SD₁₆×C₉₇	2
C₂×#124×C₉₇	2
C₂×C₄×Q₈×C₉₇	2
#464×C₉₇	2
C₂×C₄⋊C₈×C₉₇	2
C₂×#103×C₉₇	2
C₂×#102×C₉₇	2
C₂×#464	97

Subgroups — C2×#464×C97 #27138992

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#464×C₉₇ #27138992