Subgroups — #446×D₉₇ #27150659

Order 12416.

400 conjugacy classes of subgroups.

The subgroup lattice has 240 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×Q₈×D₉₇	C₂	1	1	yes	maximal, characteristic
2	#20260866	C₂	1	1	yes	maximal, characteristic, complemented
3	#20260865	C₂	1	1	yes	maximal, characteristic, complemented
4	#20260864	C₂	1	2	yes	maximal, complemented
5	#20260863	C₂	1	2	yes	maximal, complemented
6	#20260862	C₂	1	1	yes	maximal, characteristic, complemented
7	C₄⋊C₈×D₉₇	C₂	1	1	yes	maximal, characteristic
8	C₄⋊C₈×D₉₇	C₂	1	2	yes	maximal
9	#97×D₉₇	C₂	1	2	yes	maximal
10	C₄×C₈×D₉₇	C₂	1	1	yes	maximal, characteristic
11	#446×C₉₇	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×Q₈×D₉₇	C₄	1	1	yes	characteristic
13	C₂×C₈×D₉₇	C₂²	1	1	yes	characteristic
14	C₂×C₈×D₉₇	C₂²	1	2	yes	—
15	C₂×C₈×D₉₇	C₂²	1	1	yes	characteristic
16	Q₈×Dic₉₇	C₄	1	1	yes	characteristic
17	C₄×Dic₁₉₄	C₂²	1	2	yes	—
18	C₄×Dic₁₉₄	C₂²	1	1	yes	characteristic
19	#15690775	C₄	1	2	yes	—
20	#15690775	C₄	1	1	yes	characteristic
21	C₄⋊C₄×D₉₇	C₄	1	2	yes	—
22	C₄⋊C₄×D₉₇	C₄	1	1	yes	characteristic
23	C₄²×D₉₇	C₂²	1	1	yes	characteristic
24	C₄²×D₉₇	C₂²	1	2	yes	—
25	C₄⋊#11715895	C₂²	1	1	yes	characteristic
26	C₄⋊#11715895	C₂²	1	2	yes	—
27	#15690739	C₂²	1	2	yes	—
28	#15690739	C₂²	1	2	yes	—
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	1	yes	characteristic
34	C₈×Dic₉₇	C₂²	1	1	yes	characteristic
35	C₈×Dic₉₇	C₂²	1	1	yes	characteristic
36	C₄×Q₈×C₉₇	C₂²	1	1	yes	characteristic
37	C₄⋊C₈×C₉₇	C₂²	1	2	yes	—
38	C₄⋊C₈×C₉₇	C₂²	1	1	yes	characteristic
39	#97×C₉₇	C₂²	1	2	yes	—
40	C₄×C₈×C₉₇	C₂²	1	1	yes	characteristic
41	C₂×Dic₁₉₄	C₂×C₄	1	1	yes	characteristic
42	C₂×Dic₁₉₄	C₂×C₄	1	2	yes	—
43	Q₈×D₉₇	—	2	4	no	—
44	C₂×C₄×D₉₇	C₂×C₄	1	1	yes	characteristic
45	C₂×C₄×D₉₇	C₂×C₄	1	2	yes	—
46	C₂×C₄×D₉₇	C₂×C₄	1	2	yes	—
47	C₂×C₄×D₉₇	C₂×C₄	1	1	yes	characteristic
48	C₂×C₄×D₉₇	C₂³	1	1	yes	characteristic
49	C₂×#11715895	C₂³	1	1	yes	characteristic
50	C₂×#11715895	C₂³	1	2	yes	—
51	C₂×#11715895	C₂³	1	1	yes	characteristic
52	C₈×D₉₇	—	2	2	no	—
53	C₈×D₉₇	—	2	4	no	—
54	C₈×D₉₇	Q₈	1	4	yes	complemented
55	C₄⋊Dic₉₇	C₂×C₄	1	2	yes	—
56	C₄⋊Dic₉₇	C₂×C₄	1	1	yes	characteristic
57	#13095490	C₂×C₄	1	2	yes	—
58	#13095490	C₂×C₄	1	2	yes	—
59	#13095490	C₂×C₄	1	2	yes	—
60	C₄×Dic₉₇	C₂×C₄	1	2	yes	—
61	C₄×Dic₉₇	C₂×C₄	1	1	yes	characteristic
62	C₄×Dic₉₇	C₂×C₄	1	2	yes	—
63	C₄×Dic₉₇	C₂×C₄	1	1	yes	characteristic
64	C₄×Dic₉₇	C₂³	1	1	yes	characteristic
65	C₄×Dic₉₇	C₂³	1	2	yes	—
66	C₂×Q₈×C₉₇	C₂×C₄	1	1	yes	characteristic
67	C₂×C₈×C₉₇	C₂³	1	1	yes	characteristic
68	C₂×C₈×C₉₇	C₂³	1	2	yes	—
69	C₂×C₈×C₉₇	C₂³	1	1	yes	characteristic
70	C₄⋊C₄×C₉₇	C₂×C₄	1	2	yes	—
71	C₄⋊C₄×C₉₇	C₂×C₄	1	1	yes	characteristic
72	C₄²×C₉₇	C₂³	1	1	yes	characteristic
73	C₄²×C₉₇	C₂³	1	2	yes	—
74	C₂²×D₉₇	C₂²×C₄	1	1	yes	characteristic
75	Q₈×C₉₇	—	2	2	no	—
76	C₂×C₄×C₉₇	C₂²×C₄	1	1	yes	characteristic
77	C₂×C₄×C₉₇	C₂²×C₄	1	2	yes	—
78	C₂×C₄×C₉₇	C₂²×C₄	1	2	yes	—
79	C₂×C₄×C₉₇	C₂²×C₄	1	1	yes	characteristic
80	C₂×C₄×C₉₇	C₂⁴	1	1	yes	characteristic
81	C₂×Dic₉₇	C₂²×C₄	1	1	yes	characteristic
82	C₂×Dic₉₇	C₂²×C₄	1	2	yes	—
83	C₂×Dic₉₇	C₂²×C₄	1	2	yes	—
84	C₂×Dic₉₇	C₂²×C₄	1	1	yes	characteristic
85	C₂×Dic₉₇	C₂²×C₄	1	1	yes	characteristic
86	C₄×D₉₇	—	2	4	no	—
87	C₄×D₉₇	—	2	2	no	—
88	C₄×D₉₇	—	2	4	no	—
89	C₄×D₉₇	M₁₆	1	4	yes	—
90	C₄×D₉₇	C₂×Q₈	1	2	yes	—
91	C₄×D₉₇	#41	1	2	yes	—
92	Dic₁₉₄	—	2	2	no	—
93	Dic₁₉₄	—	2	4	no	—
94	C₈×C₉₇	—	2	1	no	—
95	C₈×C₉₇	—	2	2	no	—
96	C₈×C₉₇	C₂×Q₈	1	2	yes	complemented
97	#11715895	—	2	1	no	—
98	#11715895	—	2	2	no	—
99	#11715895	C₂×Q₈	1	2	yes	complemented
100	C₂²×C₉₇	C₂³×C₄	1	1	yes	characteristic
101	C₂×D₉₇	C₄×Q₈	1	2	yes	—
102	C₂×D₉₇	C₂×M₁₆	1	2	yes	—
103	C₂×D₉₇	#131	1	2	yes	—
104	C₄×C₉₇	—	2	2	no	—
105	C₄×C₉₇	—	2	1	no	—
106	C₄×C₉₇	—	2	2	no	—
107	C₄×C₉₇	C₂×M₁₆	1	2	yes	—
108	C₄×C₉₇	C₂²×Q₈	1	1	yes	characteristic
109	C₄×C₉₇	C₂×#41	1	1	yes	characteristic
110	Dic₉₇	—	2	2	no	—
111	Dic₉₇	—	2	1	no	—
112	Dic₉₇	—	2	2	no	—
113	Dic₉₇	C₄×Q₈	1	2	yes	—
114	Dic₉₇	C₂×M₁₆	1	2	yes	—
115	C₂×C₉₇	C₂×C₄×Q₈	1	1	yes	characteristic
116	C₂×C₉₇	C₂²×M₁₆	1	1	yes	characteristic
117	C₂×C₉₇	C₂×#131	1	1	yes	characteristic
118	D₉₇	#446	1	4	yes	direct product factor
119	C₂×#446	—	97	1	no	maximal, Sylow
120	C₉₇	C₂×#446	1	1	yes	characteristic, complemented, Sylow
121	C₂×C₄×Q₈	—	97	1	no	—
122	#446	—	97	1	no	—
123	#446	—	97	2	no	—
124	#446	—	97	1	no	—
125	#446	—	97	2	no	—
126	#446	—	97	1	no	—
127	#446	D₉₇	1	1	yes	characteristic, direct product factor
128	C₂×C₄⋊C₈	—	97	1	no	—
129	C₂×C₄⋊C₈	—	97	2	no	—
130	C₂×#97	—	97	2	no	—
131	C₂×C₄×C₈	—	97	1	no	—
132	C₂²×Q₈	—	97	1	no	—
133	C₂²×C₈	—	97	1	no	—
134	C₂²×C₈	—	97	1	no	—
135	C₂²×C₈	—	97	2	no	—
136	C₄×Q₈	—	97	1	no	—
137	C₄×Q₈	—	97	2	no	—
138	C₄×Q₈	—	97	1	no	—
139	C₄×Q₈	—	97	2	no	—
140	C₄×Q₈	—	97	1	no	—
141	C₄×Q₈	C₂×D₉₇	1	1	yes	characteristic
142	C₂×C₄⋊C₄	—	97	1	no	—
143	C₂×C₄⋊C₄	—	97	2	no	—
144	C₂×C₄²	—	97	1	no	—
145	C₂×C₄²	—	97	2	no	—
146	C₄⋊C₈	—	97	2	no	—
147	C₄⋊C₈	—	97	2	no	—
148	C₄⋊C₈	—	97	2	no	—
149	C₄⋊C₈	—	97	1	no	—
150	C₄⋊C₈	—	97	2	no	—
151	C₄⋊C₈	C₂×D₉₇	1	1	yes	characteristic
152	C₄⋊C₈	C₂×D₉₇	1	2	yes	—
153	#97	—	97	2	no	—
154	#97	—	97	2	no	—
155	#97	—	97	2	no	—
156	#97	C₂×D₉₇	1	2	yes	—
157	C₄×C₈	—	97	1	no	—
158	C₄×C₈	—	97	1	no	—
159	C₄×C₈	—	97	1	no	—
160	C₄×C₈	C₂×D₉₇	1	1	yes	characteristic
161	C₂×Q₈	—	97	2	no	—
162	C₂×Q₈	—	97	1	no	—
163	C₂×Q₈	—	194	4	no	—
164	C₂×Q₈	C₄×D₉₇	1	1	yes	characteristic
165	C₂²×C₄	—	97	1	no	—
166	C₂²×C₄	—	97	1	no	—
167	C₂²×C₄	—	97	1	no	—
168	C₂²×C₄	—	97	2	no	—
169	C₂²×C₄	—	97	2	no	—
170	C₂×C₈	—	97	1	no	—
171	C₂×C₈	—	97	4	no	—
172	C₂×C₈	—	97	1	no	—
173	C₂×C₈	—	97	2	no	—
174	C₂×C₈	—	194	2	no	—
175	C₂×C₈	—	194	4	no	—
176	C₂×C₈	C₂²×D₉₇	1	1	yes	characteristic
177	C₂×C₈	C₂²×D₉₇	1	2	yes	—
178	C₂×C₈	C₂²×D₉₇	1	1	yes	characteristic
179	C₄⋊C₄	—	97	2	no	—
180	C₄⋊C₄	—	97	2	no	—
181	C₄⋊C₄	—	97	2	no	—
182	C₄⋊C₄	—	97	2	no	—
183	C₄⋊C₄	—	97	1	no	—
184	C₄⋊C₄	C₄×D₉₇	1	2	yes	—
185	C₄⋊C₄	C₄×D₉₇	1	1	yes	characteristic
186	C₄²	—	97	1	no	—
187	C₄²	—	97	2	no	—
188	C₄²	—	97	1	no	—
189	C₄²	—	97	2	no	—
190	C₄²	—	97	1	no	—
191	C₄²	—	97	2	no	—
192	C₄²	C₂²×D₉₇	1	1	yes	characteristic
193	C₄²	C₂²×D₉₇	1	2	yes	—
194	C₂³	—	97	1	no	—
195	Q₈	—	2	2	no	—
196	Q₈	—	194	4	no	—
197	Q₈	—	194	2	no	—
198	C₂×C₄	—	97	1	no	—
199	C₂×C₄	—	97	1	no	—
200	C₂×C₄	—	97	4	no	—
201	C₂×C₄	—	97	2	no	—
202	C₂×C₄	—	97	2	no	—
203	C₂×C₄	—	97	1	no	—
204	C₂×C₄	—	97	2	no	—
205	C₂×C₄	—	97	2	no	—
206	C₂×C₄	—	194	4	no	—
207	C₂×C₄	—	194	2	no	—
208	C₂×C₄	—	194	4	no	—
209	C₂×C₄	C₂×C₄×D₉₇	1	1	yes	characteristic
210	C₂×C₄	C₂×C₄×D₉₇	1	2	yes	—
211	C₂×C₄	C₂×C₄×D₉₇	1	2	yes	—
212	C₂×C₄	C₂×C₄×D₉₇	1	1	yes	characteristic
213	C₂×C₄	C₂³×D₉₇	1	1	yes	characteristic
214	C₈	—	2	1	no	—
215	C₈	—	2	2	no	—
216	C₈	—	97	2	no	—
217	C₈	—	194	1	no	—
218	C₈	—	194	2	no	—
219	C₈	Q₈×D₉₇	1	2	yes	complemented
220	C₂²	—	97	2	no	—
221	C₂²	—	97	2	no	—
222	C₂²	—	97	2	no	—
223	C₂²	C₂²×C₄×D₉₇	1	1	yes	characteristic
224	C₄	—	2	2	no	—
225	C₄	—	2	1	no	—
226	C₄	—	2	2	no	—
227	C₄	—	97	2	no	—
228	C₄	—	97	2	no	—
229	C₄	—	194	2	no	—
230	C₄	—	194	1	no	—
231	C₄	—	194	2	no	—
232	C₄	M₁₆×D₉₇	1	2	yes	—
233	C₄	C₂×Q₈×D₉₇	1	1	yes	characteristic
234	C₄	#41×D₉₇	1	1	yes	characteristic
235	C₂	—	97	4	no	—
236	C₂	C₄×Q₈×D₉₇	1	1	yes	characteristic
237	C₂	C₂×M₁₆×D₉₇	1	1	yes	characteristic
238	C₂	#131×D₉₇	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₉₇
commutator, socle	C₂²×C₉₇	C₂³×C₄
Fitting	#446×C₉₇	C₂
prime radical	C₂²×D₉₇	C₂²×C₄
cosocle	C₂×C₄×C₉₇	C₂⁴
hypercenter	#446	D₉₇
hypocenter	C₉₇	C₂×#446
LCS	G ▷ C₂²×C₉₇ ▷ C₉₇
UCS	C₁ ◁ C₂×C₄ ◁ #446

Complements

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

N	H ≅ G/N	type
2. #20260866	C₂	semidirect
3. #20260865	C₂	semidirect
4. #20260864	C₂	semidirect
5. #20260863	C₂	semidirect
6. #20260862	C₂	semidirect
11. #446×C₉₇	C₂	semidirect
54. C₈×D₉₇	Q₈	semidirect
96. C₈×C₉₇	C₂×Q₈	semidirect
99. #11715895	C₂×Q₈	semidirect
118. D₉₇	#446	direct (N × H)
120. C₉₇	C₂×#446	semidirect
127. #446	D₉₇	direct (N × H)
219. C₈	Q₈×D₉₇	semidirect

Maximal subgroups

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

subgroup	index
C₄×Q₈×D₉₇	2
#20260866	2
#20260865	2
#20260864	2
#20260863	2
#20260862	2
C₄⋊C₈×D₉₇	2
#97×D₉₇	2
C₄×C₈×D₉₇	2
#446×C₉₇	2
C₂×#446	97

Subgroups — #446×D97 #27150659

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #446×D₉₇ #27150659