Subgroups — C₉₇×#3397 #27139355

Order 12416.

406 conjugacy classes of subgroups.

The subgroup lattice has 266 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#556×C₉₇	C₂	1	1	yes	maximal, characteristic, complemented
2	#550×C₉₇	C₂	1	1	yes	maximal, characteristic, complemented
3	#496×C₉₇	C₂	1	2	yes	maximal, complemented
4	#492×C₉₇	C₂	1	1	yes	maximal, characteristic, complemented
5	#463×C₉₇	C₂	1	2	yes	maximal, complemented
6	#461×C₉₇	C₂	1	1	yes	maximal, characteristic, complemented
7	#460×C₉₇	C₂	1	2	yes	maximal, complemented
8	#459×C₉₇	C₂	1	1	yes	maximal, characteristic, complemented
9	#446×C₉₇	C₂	1	1	yes	maximal, characteristic, complemented
10	#440×C₉₇	C₂	1	2	yes	maximal, complemented
11	C₄×D₈×C₉₇	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#41×C₉₇	—	2	1	no	—
13	C₂×SD₁₆×C₉₇	C₂²	1	2	yes	—
14	C₂×SD₁₆×C₉₇	C₂²	1	2	yes	—
15	C₂×SD₁₆×C₉₇	C₂²	1	2	yes	—
16	C₂×D₈×C₉₇	C₂²	1	1	yes	characteristic
17	C₂×D₈×C₉₇	C₂²	1	2	yes	—
18	C₄⋊Q₈×C₉₇	C₂²	1	1	yes	characteristic
19	#127×C₉₇	C₂²	1	1	yes	characteristic
20	#127×C₉₇	C₂²	1	2	yes	—
21	#126×C₉₇	—	2	2	no	—
22	#124×C₉₇	C₂²	1	2	yes	—
23	C₂²⋊Q₈×C₉₇	—	2	1	no	—
24	#121×C₉₇	—	2	1	no	—
25	#121×C₉₇	—	2	2	no	—
26	C₄×Q₈×C₉₇	C₂²	1	2	yes	—
27	C₄×Q₈×C₉₇	C₂²	1	1	yes	characteristic
28	C₄×D₄×C₉₇	C₂²	1	1	yes	characteristic
29	C₄×D₄×C₉₇	C₂²	1	2	yes	—
30	C₄×D₄×C₉₇	C₂²	1	1	yes	characteristic
31	#117×C₉₇	—	2	1	no	—
32	#107×C₉₇	C₂²	1	1	yes	characteristic
33	C₄⋊C₈×C₉₇	C₂²	1	1	yes	characteristic
34	C₄⋊C₈×C₉₇	C₂²	1	2	yes	—
35	#103×C₉₇	C₂²	1	2	yes	—
36	#102×C₉₇	C₂²	1	1	yes	characteristic
37	#102×C₉₇	C₂²	1	1	yes	characteristic
38	#102×C₉₇	C₂²	1	2	yes	—
39	#97×C₉₇	C₂²	1	2	yes	—
40	C₄×C₈×C₉₇	C₂²	1	1	yes	characteristic
41	#41×C₉₇	—	4	4	no	—
42	C₂×Q₈×C₉₇	D₄	1	1	yes	characteristic
43	C₂×Q₈×C₉₇	C₂³	1	2	yes	—
44	C₂×D₄×C₉₇	—	2	2	no	—
45	C₂×D₄×C₉₇	—	2	1	no	—
46	C₂×D₄×C₉₇	—	2	2	no	—
47	C₂×D₄×C₉₇	—	2	1	no	—
48	C₂×D₄×C₉₇	C₂³	1	1	yes	characteristic
49	C₂×D₄×C₉₇	C₂³	1	1	yes	characteristic
50	C₂×D₄×C₉₇	C₂³	1	2	yes	—
51	C₂²×C₄×C₉₇	—	2	1	no	—
52	C₂²×C₄×C₉₇	—	2	2	no	—
53	C₂²×C₄×C₉₇	—	2	1	no	—
54	SD₁₆×C₉₇	—	2	4	no	—
55	SD₁₆×C₉₇	—	4	2	no	—
56	SD₁₆×C₉₇	—	4	2	no	—
57	D₈×C₉₇	—	2	2	no	—
58	D₈×C₉₇	—	4	2	no	—
59	C₂×C₈×C₉₇	C₂³	1	1	yes	characteristic
60	C₂×C₈×C₉₇	C₂³	1	2	yes	—
61	C₂×C₈×C₉₇	C₂³	1	1	yes	characteristic
62	C₄⋊C₄×C₉₇	—	2	1	no	—
63	C₄⋊C₄×C₉₇	—	2	2	no	—
64	C₄⋊C₄×C₉₇	—	2	1	no	—
65	C₄⋊C₄×C₉₇	D₄	1	1	yes	characteristic
66	C₄⋊C₄×C₉₇	D₄	1	2	yes	—
67	C₄⋊C₄×C₉₇	C₂³	1	1	yes	characteristic
68	C₄⋊C₄×C₉₇	C₂³	1	1	yes	characteristic
69	C₂²⋊C₄×C₉₇	—	2	2	no	—
70	C₂²⋊C₄×C₉₇	—	2	2	no	—
71	C₂²⋊C₄×C₉₇	—	2	1	no	—
72	C₂²⋊C₄×C₉₇	—	2	1	no	—
73	C₂²⋊C₄×C₉₇	—	2	2	no	—
74	C₄²×C₉₇	—	2	2	no	—
75	C₄²×C₉₇	C₂³	1	1	yes	characteristic
76	C₄²×C₉₇	C₂³	1	2	yes	—
77	C₂³×C₉₇	—	2	1	no	—
78	C₂³×C₉₇	—	2	1	no	—
79	C₂³×C₉₇	—	2	2	no	—
80	Q₈×C₉₇	—	2	2	no	—
81	Q₈×C₉₇	—	2	2	no	—
82	Q₈×C₉₇	—	2	2	no	—
83	D₄×C₉₇	—	2	1	no	—
84	D₄×C₉₇	—	2	1	no	—
85	D₄×C₉₇	—	2	2	no	—
86	D₄×C₉₇	—	2	1	no	—
87	D₄×C₉₇	—	2	2	no	—
88	D₄×C₉₇	—	2	1	no	—
89	D₄×C₉₇	—	4	4	no	—
90	D₄×C₉₇	—	4	2	no	—
91	D₄×C₉₇	—	8	2	no	—
92	D₄×C₉₇	—	8	1	no	—
93	C₂×C₄×C₉₇	—	2	1	no	—
94	C₂×C₄×C₉₇	—	2	2	no	—
95	C₂×C₄×C₉₇	—	2	1	no	—
96	C₂×C₄×C₉₇	—	4	4	no	—
97	C₂×C₄×C₉₇	—	4	2	no	—
98	C₂×C₄×C₉₇	—	4	2	no	—
99	C₂×C₄×C₉₇	C₂×D₄	1	2	yes	—
100	C₂×C₄×C₉₇	C₂×D₄	1	1	yes	characteristic
101	C₂×C₄×C₉₇	C₂×D₄	1	2	yes	—
102	C₂×C₄×C₉₇	C₂×D₄	1	1	yes	characteristic
103	C₂×C₄×C₉₇	C₂⁴	1	1	yes	characteristic
104	C₈×C₉₇	—	2	2	no	—
105	C₈×C₉₇	—	2	1	no	—
106	C₈×C₉₇	#41	1	2	yes	complemented
107	C₂²×C₉₇	—	4	1	no	—
108	C₂²×C₉₇	—	4	1	no	—
109	C₂²×C₉₇	—	4	2	no	—
110	C₂²×C₉₇	—	4	2	no	—
111	C₂²×C₉₇	—	4	1	no	—
112	C₂²×C₉₇	—	4	1	no	—
113	C₂²×C₉₇	—	4	1	no	—
114	C₂²×C₉₇	—	4	2	no	—
115	C₂²×C₉₇	—	4	1	no	—
116	C₂²×C₉₇	C₂²×D₄	1	1	yes	characteristic
117	C₄×C₉₇	—	2	2	no	—
118	C₄×C₉₇	—	2	2	no	—
119	C₄×C₉₇	—	2	1	no	—
120	C₄×C₉₇	—	4	2	no	—
121	C₄×C₉₇	—	4	1	no	—
122	C₄×C₉₇	—	4	1	no	—
123	C₄×C₉₇	AGL(1,ℤ/8)	1	2	yes	—
124	C₄×C₉₇	C₂×#41	1	1	yes	characteristic
125	C₄×C₉₇	H(5,𝔽₂)	1	1	yes	characteristic
126	C₂×C₉₇	—	8	1	no	—
127	C₂×C₉₇	—	8	2	no	—
128	C₂×C₉₇	—	8	1	no	—
129	C₂×C₉₇	#550	1	1	yes	characteristic
130	C₂×C₉₇	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
131	C₂×C₉₇	#577	1	1	yes	characteristic
132	#3397	C₉₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
133	C₉₇	#3397	1	1	yes	characteristic, direct product factor, Sylow
134	#556	C₂×C₉₇	1	1	yes	characteristic, complemented
135	#550	C₂×C₉₇	1	1	yes	characteristic, complemented
136	#496	C₂×C₉₇	1	2	yes	complemented
137	#492	C₂×C₉₇	1	1	yes	characteristic, complemented
138	#463	C₂×C₉₇	1	2	yes	complemented
139	#461	C₂×C₉₇	1	1	yes	characteristic, complemented
140	#460	C₂×C₉₇	1	2	yes	complemented
141	#459	C₂×C₉₇	1	1	yes	characteristic, complemented
142	#446	C₂×C₉₇	1	1	yes	characteristic, complemented
143	#440	C₂×C₉₇	1	2	yes	complemented
144	C₄×D₈	C₂×C₉₇	1	1	yes	characteristic, complemented
145	C₂×#41	—	2	1	no	—
146	C₂×SD₁₆	C₂²×C₉₇	1	2	yes	—
147	C₂×SD₁₆	C₂²×C₉₇	1	2	yes	—
148	C₂×SD₁₆	C₂²×C₉₇	1	2	yes	—
149	C₂×D₈	C₂²×C₉₇	1	1	yes	characteristic
150	C₂×D₈	C₂²×C₉₇	1	2	yes	—
151	C₄⋊Q₈	C₂²×C₉₇	1	1	yes	characteristic
152	#127	C₂²×C₉₇	1	1	yes	characteristic
153	#127	C₂²×C₉₇	1	2	yes	—
154	#126	—	2	2	no	—
155	#124	C₂²×C₉₇	1	2	yes	—
156	C₂²⋊Q₈	—	2	1	no	—
157	#121	—	2	1	no	—
158	#121	—	2	2	no	—
159	C₄×Q₈	C₂²×C₉₇	1	2	yes	—
160	C₄×Q₈	C₂²×C₉₇	1	1	yes	characteristic
161	C₄×D₄	C₂²×C₉₇	1	1	yes	characteristic
162	C₄×D₄	C₂²×C₉₇	1	2	yes	—
163	C₄×D₄	C₂²×C₉₇	1	1	yes	characteristic
164	#117	—	2	1	no	—
165	#107	C₂²×C₉₇	1	1	yes	characteristic
166	C₄⋊C₈	C₂²×C₉₇	1	1	yes	characteristic
167	C₄⋊C₈	C₂²×C₉₇	1	2	yes	—
168	#103	C₂²×C₉₇	1	2	yes	—
169	#102	C₂²×C₉₇	1	1	yes	characteristic
170	#102	C₂²×C₉₇	1	1	yes	characteristic
171	#102	C₂²×C₉₇	1	2	yes	—
172	#97	C₂²×C₉₇	1	2	yes	—
173	C₄×C₈	C₂²×C₉₇	1	1	yes	characteristic
174	#41	—	4	4	no	—
175	C₂×Q₈	D₄×C₉₇	1	1	yes	characteristic
176	C₂×Q₈	C₂³×C₉₇	1	2	yes	—
177	C₂×D₄	—	2	2	no	—
178	C₂×D₄	—	2	1	no	—
179	C₂×D₄	—	2	2	no	—
180	C₂×D₄	—	2	1	no	—
181	C₂×D₄	C₂³×C₉₇	1	1	yes	characteristic
182	C₂×D₄	C₂³×C₉₇	1	1	yes	characteristic
183	C₂×D₄	C₂³×C₉₇	1	2	yes	—
184	C₂²×C₄	—	2	1	no	—
185	C₂²×C₄	—	2	2	no	—
186	C₂²×C₄	—	2	1	no	—
187	SD₁₆	—	2	4	no	—
188	SD₁₆	—	4	2	no	—
189	SD₁₆	—	4	2	no	—
190	D₈	—	2	2	no	—
191	D₈	—	4	2	no	—
192	C₂×C₈	C₂³×C₉₇	1	1	yes	characteristic
193	C₂×C₈	C₂³×C₉₇	1	2	yes	—
194	C₂×C₈	C₂³×C₉₇	1	1	yes	characteristic
195	C₄⋊C₄	—	2	1	no	—
196	C₄⋊C₄	—	2	2	no	—
197	C₄⋊C₄	—	2	1	no	—
198	C₄⋊C₄	D₄×C₉₇	1	1	yes	characteristic
199	C₄⋊C₄	D₄×C₉₇	1	2	yes	—
200	C₄⋊C₄	C₂³×C₉₇	1	1	yes	characteristic
201	C₄⋊C₄	C₂³×C₉₇	1	1	yes	characteristic
202	C₂²⋊C₄	—	2	2	no	—
203	C₂²⋊C₄	—	2	2	no	—
204	C₂²⋊C₄	—	2	1	no	—
205	C₂²⋊C₄	—	2	1	no	—
206	C₂²⋊C₄	—	2	2	no	—
207	C₄²	—	2	2	no	—
208	C₄²	C₂³×C₉₇	1	1	yes	characteristic
209	C₄²	C₂³×C₉₇	1	2	yes	—
210	C₂³	—	2	1	no	—
211	C₂³	—	2	1	no	—
212	C₂³	—	2	2	no	—
213	Q₈	—	2	2	no	—
214	Q₈	—	2	2	no	—
215	Q₈	—	2	2	no	—
216	D₄	—	2	1	no	—
217	D₄	—	2	1	no	—
218	D₄	—	2	2	no	—
219	D₄	—	2	1	no	—
220	D₄	—	2	2	no	—
221	D₄	—	2	1	no	—
222	D₄	—	4	4	no	—
223	D₄	—	4	2	no	—
224	D₄	—	8	2	no	—
225	D₄	—	8	1	no	—
226	C₂×C₄	—	2	1	no	—
227	C₂×C₄	—	2	2	no	—
228	C₂×C₄	—	2	1	no	—
229	C₂×C₄	—	4	4	no	—
230	C₂×C₄	—	4	2	no	—
231	C₂×C₄	—	4	2	no	—
232	C₂×C₄	C₂×D₄×C₉₇	1	2	yes	—
233	C₂×C₄	C₂×D₄×C₉₇	1	1	yes	characteristic
234	C₂×C₄	C₂×D₄×C₉₇	1	2	yes	—
235	C₂×C₄	C₂×D₄×C₉₇	1	1	yes	characteristic
236	C₂×C₄	C₂⁴×C₉₇	1	1	yes	characteristic
237	C₈	—	2	2	no	—
238	C₈	—	2	1	no	—
239	C₈	#41×C₉₇	1	2	yes	complemented
240	C₂²	—	4	1	no	—
241	C₂²	—	4	1	no	—
242	C₂²	—	4	2	no	—
243	C₂²	—	4	2	no	—
244	C₂²	—	4	1	no	—
245	C₂²	—	4	1	no	—
246	C₂²	—	4	1	no	—
247	C₂²	—	4	2	no	—
248	C₂²	—	4	1	no	—
249	C₂²	C₂²×D₄×C₉₇	1	1	yes	characteristic
250	C₄	—	2	2	no	—
251	C₄	—	2	2	no	—
252	C₄	—	2	1	no	—
253	C₄	—	4	2	no	—
254	C₄	—	4	1	no	—
255	C₄	—	4	1	no	—
256	C₄	AGL(1,ℤ/8)×C₉₇	1	2	yes	—
257	C₄	C₂×#41×C₉₇	1	1	yes	characteristic
258	C₄	H(5,𝔽₂)×C₉₇	1	1	yes	characteristic
259	C₂	—	8	1	no	—
260	C₂	—	8	2	no	—
261	C₂	—	8	1	no	—
262	C₂	#550×C₉₇	1	1	yes	characteristic
263	C₂	C₂×AGL(1,ℤ/8)×C₉₇	1	1	yes	characteristic
264	C₂	#577×C₉₇	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₉₇	C₂²×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁴×C₉₇
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₉₇ ◁ C₄×Q₈×C₉₇ ◁ G

Complements

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

N	H ≅ G/N	type
1. #556×C₉₇	C₂	semidirect
2. #550×C₉₇	C₂	semidirect
3. #496×C₉₇	C₂	semidirect
4. #492×C₉₇	C₂	semidirect
5. #463×C₉₇	C₂	semidirect
6. #461×C₉₇	C₂	semidirect
7. #460×C₉₇	C₂	semidirect
8. #459×C₉₇	C₂	semidirect
9. #446×C₉₇	C₂	semidirect
10. #440×C₉₇	C₂	semidirect
11. C₄×D₈×C₉₇	C₂	semidirect
106. C₈×C₉₇	#41	semidirect
132. #3397	C₉₇	direct (N × H)
133. C₉₇	#3397	direct (N × H)
134. #556	C₂×C₉₇	semidirect
135. #550	C₂×C₉₇	semidirect
136. #496	C₂×C₉₇	semidirect
137. #492	C₂×C₉₇	semidirect
138. #463	C₂×C₉₇	semidirect
139. #461	C₂×C₉₇	semidirect
140. #460	C₂×C₉₇	semidirect
141. #459	C₂×C₉₇	semidirect
142. #446	C₂×C₉₇	semidirect
143. #440	C₂×C₉₇	semidirect
144. C₄×D₈	C₂×C₉₇	semidirect
239. C₈	#41×C₉₇	semidirect

Maximal subgroups

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

subgroup	index
#556×C₉₇	2
#550×C₉₇	2
#496×C₉₇	2
#492×C₉₇	2
#463×C₉₇	2
#461×C₉₇	2
#460×C₉₇	2
#459×C₉₇	2
#446×C₉₇	2
#440×C₉₇	2
C₄×D₈×C₉₇	2
#3397	97

Subgroups — C97×#3397 #27139355

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₉₇×#3397 #27139355