Subgroups — #27150682

Order 12416.

356 conjugacy classes of subgroups.

The subgroup lattice has 234 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20261288	C₂	1	1	yes	maximal, characteristic
2	#20260864	C₂	1	2	yes	maximal, complemented
3	#20260864	C₂	1	2	yes	maximal, complemented
4	#20260862	C₂	1	2	yes	maximal, complemented
5	#20260859	C₂	1	2	yes	maximal, complemented
6	#20260763	C₂	1	1	yes	maximal, characteristic
7	C₂×#15690739	C₂	1	1	yes	maximal, characteristic
8	#20260691	C₂	1	1	yes	maximal, characteristic
9	#20260690	C₂	1	1	yes	maximal, characteristic
10	M₁₆×Dic₉₇	C₂	1	1	yes	maximal, characteristic
11	#424×C₉₇	C₂	1	1	yes	maximal, characteristic
12	C₂×#13095495	C₂²	1	1	yes	characteristic
13	C₂²×#11715895	C₂²	1	1	yes	characteristic
14	#15690809	C₄	1	1	yes	characteristic
15	#15690809	C₄	1	1	yes	characteristic
16	#15690801	C₄	1	1	yes	characteristic
17	#15690801	C₄	1	1	yes	characteristic
18	#15690787	C₄	1	2	yes	—
19	#15690787	C₄	1	2	yes	—
20	C₄×Dic₁₉₄	C₂²	1	2	yes	—
21	C₄×Dic₁₉₄	C₂²	1	2	yes	—
22	#15690769	C₂²	1	1	yes	characteristic
23	C₂×C₄×Dic₉₇	C₂²	1	1	yes	characteristic
24	#15690739	C₂²	1	1	yes	characteristic
25	#15690739	C₂²	1	1	yes	characteristic
26	#15690739	C₂²	1	2	yes	—
27	#15690739	C₂²	1	1	yes	characteristic
28	#15690739	C₂²	1	1	yes	characteristic
29	#15690739	C₂²	1	2	yes	—
30	#15690721	C₂²	1	2	yes	—
31	#15690720	C₂²	1	2	yes	—
32	#15690720	C₂²	1	2	yes	—
33	C₄×#11715895	C₂²	1	2	yes	—
34	C₈×Dic₉₇	C₂²	1	2	yes	—
35	C₈×Dic₉₇	C₂²	1	2	yes	—
36	C₂×M₁₆×C₉₇	C₂²	1	1	yes	characteristic
37	C₂²×C₈×C₉₇	C₂²	1	1	yes	characteristic
38	#117×C₉₇	C₂²	1	1	yes	characteristic
39	C₄⋊C₈×C₉₇	C₂²	1	2	yes	—
40	C₄⋊C₈×C₉₇	C₂²	1	2	yes	—
41	C₂×Dic₁₉₄	C₂×C₄	1	2	yes	—
42	C₂×Dic₁₉₄	C₂×C₄	1	2	yes	—
43	C₂²×Dic₉₇	C₂×C₄	1	1	yes	characteristic
44	C₂²×Dic₉₇	C₂×C₄	1	1	yes	characteristic
45	#13095495	Q₈	1	4	yes	complemented
46	C₂×#11715895	—	2	2	no	—
47	C₂×#11715895	C₂³	1	1	yes	characteristic
48	C₂×#11715895	C₂³	1	1	yes	characteristic
49	C₂×#11715895	C₂³	1	2	yes	—
50	C₄⋊Dic₉₇	C₂×C₄	1	1	yes	characteristic
51	C₄⋊Dic₉₇	C₂×C₄	1	1	yes	characteristic
52	C₄⋊Dic₉₇	C₂×C₄	1	2	yes	—
53	C₄⋊Dic₉₇	C₂×C₄	1	2	yes	—
54	#13095490	C₂×C₄	1	2	yes	—
55	#13095490	C₂×C₄	1	2	yes	—
56	#13095490	C₂×C₄	1	2	yes	—
57	#13095490	C₂×C₄	1	2	yes	—
58	C₂²⋊Dic₉₇	C₂×C₄	1	2	yes	—
59	C₄×Dic₉₇	C₂×C₄	1	1	yes	characteristic
60	C₄×Dic₉₇	C₂×C₄	1	1	yes	characteristic
61	C₄×Dic₉₇	C₂³	1	1	yes	characteristic
62	C₄×Dic₉₇	C₂³	1	1	yes	characteristic
63	C₄×Dic₉₇	C₂³	1	2	yes	—
64	C₂²×C₄×C₉₇	C₂³	1	1	yes	characteristic
65	M₁₆×C₉₇	—	2	2	no	—
66	C₂×C₈×C₉₇	—	2	2	no	—
67	C₂×C₈×C₉₇	C₂³	1	1	yes	characteristic
68	C₂×C₈×C₉₇	C₂³	1	1	yes	characteristic
69	C₂×C₈×C₉₇	C₂³	1	2	yes	—
70	C₄⋊C₄×C₉₇	C₂×C₄	1	1	yes	characteristic
71	C₄⋊C₄×C₉₇	C₂×C₄	1	1	yes	characteristic
72	C₂²⋊C₄×C₉₇	C₂×C₄	1	2	yes	—
73	C₄²×C₉₇	C₂³	1	2	yes	—
74	C₂³×C₉₇	C₂²×C₄	1	1	yes	characteristic
75	C₂×C₄×C₉₇	C₂²×C₄	1	2	yes	—
76	C₂×C₄×C₉₇	C₂²×C₄	1	2	yes	—
77	C₂×C₄×C₉₇	C₂²×C₄	1	1	yes	characteristic
78	C₂×C₄×C₉₇	C₂×Q₈	1	2	yes	—
79	C₂×C₄×C₉₇	#41	1	2	yes	—
80	C₂×C₄×C₉₇	C₂⁴	1	1	yes	characteristic
81	C₂×Dic₉₇	—	2	2	no	—
82	C₂×Dic₉₇	—	4	1	no	—
83	C₂×Dic₉₇	C₂²×C₄	1	1	yes	characteristic
84	C₂×Dic₉₇	C₂²×C₄	1	1	yes	characteristic
85	C₂×Dic₉₇	C₂²×C₄	1	1	yes	characteristic
86	C₂×Dic₉₇	C₂²×C₄	1	2	yes	—
87	C₂×Dic₉₇	C₂²×C₄	1	2	yes	—
88	C₂×Dic₉₇	C₂²×C₄	1	1	yes	characteristic
89	Dic₁₉₄	—	2	4	no	—
90	Dic₁₉₄	—	4	2	no	—
91	C₈×C₉₇	—	2	1	no	—
92	C₈×C₉₇	—	2	1	no	—
93	C₈×C₉₇	—	2	2	no	—
94	#11715895	—	2	1	no	—
95	#11715895	—	2	1	no	—
96	#11715895	C₂×Q₈	1	4	yes	—
97	C₂²×C₉₇	—	2	1	no	—
98	C₂²×C₉₇	—	2	1	no	—
99	C₂²×C₉₇	C₄×Q₈	1	2	yes	—
100	C₂²×C₉₇	C₂³×C₄	1	1	yes	characteristic
101	C₄×C₉₇	—	2	2	no	—
102	C₄×C₉₇	—	2	2	no	—
103	C₄×C₉₇	C₄×Q₈	1	2	yes	—
104	C₄×C₉₇	C₂²×Q₈	1	1	yes	characteristic
105	C₄×C₉₇	C₂×#41	1	1	yes	characteristic
106	Dic₉₇	—	2	2	no	—
107	Dic₉₇	—	2	1	no	—
108	Dic₉₇	—	2	1	no	—
109	Dic₉₇	—	2	2	no	—
110	Dic₉₇	—	2	1	no	—
111	Dic₉₇	#131	1	2	yes	—
112	C₂×C₉₇	—	2	2	no	—
113	C₂×C₉₇	C₂×C₄×Q₈	1	1	yes	characteristic
114	C₂×C₉₇	C₂×#131	1	1	yes	characteristic
115	C₂×C₉₇	#568	1	1	yes	characteristic
116	#2962	—	97	1	no	maximal, Sylow
117	C₉₇	#2962	1	1	yes	characteristic, complemented, Sylow
118	#533	—	97	1	no	—
119	#446	—	97	2	no	—
120	#446	—	97	2	no	—
121	#446	—	97	2	no	—
122	C₈×Q₈	—	97	2	no	—
123	#424	—	97	1	no	—
124	#424	D₉₇	1	1	yes	characteristic
125	C₂×C₄⋊C₈	—	97	1	no	—
126	#405	—	97	1	no	—
127	#405	—	97	1	no	—
128	C₄×M₁₆	—	97	1	no	—
129	C₂×M₁₆	—	97	1	no	—
130	C₂×M₁₆	C₂×D₉₇	1	1	yes	characteristic
131	C₂²×C₈	—	97	1	no	—
132	C₂²×C₈	C₂×D₉₇	1	1	yes	characteristic
133	C₄⋊Q₈	—	97	1	no	—
134	C₄⋊Q₈	—	97	1	no	—
135	#125	—	97	1	no	—
136	#125	—	97	1	no	—
137	C₂²⋊Q₈	—	97	2	no	—
138	C₂²⋊Q₈	—	97	2	no	—
139	C₄×Q₈	—	97	2	no	—
140	C₄×Q₈	—	97	2	no	—
141	#117	—	97	1	no	—
142	#117	C₂×D₉₇	1	1	yes	characteristic
143	C₂×C₄²	—	97	1	no	—
144	C₄⋊C₈	—	97	1	no	—
145	C₄⋊C₈	—	97	1	no	—
146	C₄⋊C₈	—	97	1	no	—
147	C₄⋊C₈	—	97	1	no	—
148	C₄⋊C₈	—	97	2	no	—
149	C₄⋊C₈	—	97	2	no	—
150	C₄⋊C₈	C₂×D₉₇	1	2	yes	—
151	C₄⋊C₈	C₂×D₉₇	1	2	yes	—
152	#97	—	97	2	no	—
153	#97	—	97	2	no	—
154	#97	—	97	2	no	—
155	C₄×C₈	—	97	2	no	—
156	C₄×C₈	—	97	2	no	—
157	C₄×C₈	—	97	2	no	—
158	C₂×Q₈	—	97	2	no	—
159	C₂×Q₈	—	97	2	no	—
160	C₂²×C₄	—	97	1	no	—
161	C₂²×C₄	—	97	1	no	—
162	C₂²×C₄	C₂²×D₉₇	1	1	yes	characteristic
163	M₁₆	—	2	2	no	—
164	M₁₆	—	97	4	no	—
165	C₂×C₈	—	2	2	no	—
166	C₂×C₈	—	97	2	no	—
167	C₂×C₈	—	97	1	no	—
168	C₂×C₈	—	97	1	no	—
169	C₂×C₈	—	194	2	no	—
170	C₂×C₈	C₂²×D₉₇	1	1	yes	characteristic
171	C₂×C₈	C₂²×D₉₇	1	2	yes	—
172	C₂×C₈	C₂²×D₉₇	1	1	yes	characteristic
173	C₄⋊C₄	—	97	2	no	—
174	C₄⋊C₄	—	97	2	no	—
175	C₄⋊C₄	—	97	2	no	—
176	C₄⋊C₄	—	97	2	no	—
177	C₄⋊C₄	—	97	1	no	—
178	C₄⋊C₄	—	97	1	no	—
179	C₄⋊C₄	—	97	2	no	—
180	C₄⋊C₄	—	97	2	no	—
181	C₄⋊C₄	C₄×D₉₇	1	1	yes	characteristic
182	C₄⋊C₄	C₄×D₉₇	1	1	yes	characteristic
183	C₂²⋊C₄	—	97	2	no	—
184	C₂²⋊C₄	C₄×D₉₇	1	2	yes	—
185	C₄²	—	97	1	no	—
186	C₄²	—	97	1	no	—
187	C₄²	—	97	1	no	—
188	C₄²	—	97	2	no	—
189	C₄²	—	97	1	no	—
190	C₄²	C₂²×D₉₇	1	2	yes	—
191	C₂³	C₂×C₄×D₉₇	1	1	yes	characteristic
192	Q₈	—	388	2	no	—
193	Q₈	—	194	4	no	—
194	C₂×C₄	—	97	1	no	—
195	C₂×C₄	—	97	1	no	—
196	C₂×C₄	—	97	1	no	—
197	C₂×C₄	—	97	1	no	—
198	C₂×C₄	—	97	2	no	—
199	C₂×C₄	—	97	2	no	—
200	C₂×C₄	—	388	1	no	—
201	C₂×C₄	—	194	2	no	—
202	C₂×C₄	C₂×C₄×D₉₇	1	2	yes	—
203	C₂×C₄	C₂×C₄×D₉₇	1	2	yes	—
204	C₂×C₄	C₂×C₄×D₉₇	1	1	yes	characteristic
205	C₂×C₄	Q₈×D₉₇	1	2	yes	—
206	C₂×C₄	#13095510	1	2	yes	—
207	C₂×C₄	C₂³×D₉₇	1	1	yes	characteristic
208	C₈	—	2	1	no	—
209	C₈	—	2	1	no	—
210	C₈	—	2	2	no	—
211	C₈	—	97	4	no	—
212	C₈	—	194	1	no	—
213	C₈	—	194	1	no	—
214	C₂²	—	2	1	no	—
215	C₂²	—	2	1	no	—
216	C₂²	#15690775	1	2	yes	—
217	C₂²	C₂²×C₄×D₉₇	1	1	yes	characteristic
218	C₄	—	2	2	no	—
219	C₄	—	2	2	no	—
220	C₄	—	97	2	no	—
221	C₄	—	194	1	no	—
222	C₄	—	194	2	no	—
223	C₄	—	194	1	no	—
224	C₄	—	194	1	no	—
225	C₄	—	194	2	no	—
226	C₄	#15690775	1	2	yes	—
227	C₄	C₂×Q₈×D₉₇	1	1	yes	characteristic
228	C₄	C₂×#13095510	1	1	yes	characteristic
229	C₂	—	2	2	no	—
230	C₂	C₂×#15690775	1	1	yes	characteristic
231	C₂	#131×D₉₇	1	1	yes	characteristic
232	C₂	#20261500	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₉₇
commutator, socle	C₂²×C₉₇	C₂³×C₄
Fitting	#424×C₉₇	C₂
prime radical	C₂³×C₉₇	C₂²×C₄
cosocle	C₂×C₄×C₉₇	C₂⁴
hypercenter	#424	D₉₇
hypocenter	C₉₇	#2962
LCS	G ▷ C₂²×C₉₇ ▷ C₉₇
UCS	C₁ ◁ C₂×C₄ ◁ #424

Complements

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

N	H ≅ G/N	type
2. #20260864	C₂	semidirect
3. #20260864	C₂	semidirect
4. #20260862	C₂	semidirect
5. #20260859	C₂	semidirect
45. #13095495	Q₈	semidirect
117. C₉₇	#2962	semidirect

Maximal subgroups

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

subgroup	index
#20261288	2
#20260864	2
#20260862	2
#20260859	2
#20260763	2
C₂×#15690739	2
#20260691	2
#20260690	2
M₁₆×Dic₉₇	2
#424×C₉₇	2
#2962	97