Subgroups — C₉₇×#3284 #27139242

Order 12416.

494 conjugacy classes of subgroups.

The subgroup lattice has 282 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#135×C₉₇	C₂	1	2	yes	maximal, complemented
2	#550×C₉₇	C₂	1	1	yes	maximal, characteristic, complemented
3	#547×C₉₇	C₂	1	1	yes	maximal, characteristic, complemented
4	#492×C₉₇	C₂	1	1	yes	maximal, characteristic, complemented
5	#465×C₉₇	C₂	1	2	yes	maximal, complemented
6	#462×C₉₇	C₂	1	1	yes	maximal, characteristic, complemented
7	#459×C₉₇	C₂	1	1	yes	maximal, characteristic, complemented
8	#452×C₉₇	C₂	1	2	yes	maximal, complemented
9	#449×C₉₇	C₂	1	2	yes	maximal, complemented
10	C₄×SD₁₆×C₉₇	C₂	1	1	yes	maximal, characteristic, complemented
11	C₈×D₄×C₉₇	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#41×C₉₇	C₂²	1	2	yes	—
13	C₂×#41×C₉₇	C₂²	1	2	yes	—
14	#135×C₉₇	—	2	8	no	—
15	C₂×Q₁₆×C₉₇	C₂²	1	2	yes	—
16	C₂×SD₁₆×C₉₇	C₂²	1	2	yes	—
17	C₂×SD₁₆×C₉₇	C₂²	1	2	yes	—
18	C₂×SD₁₆×C₉₇	C₂²	1	1	yes	characteristic
19	C₂×D₈×C₉₇	C₂²	1	2	yes	—
20	C₂²×C₈×C₉₇	C₂²	1	2	yes	—
21	C₄⋊Q₈×C₉₇	C₂²	1	1	yes	characteristic
22	#127×C₉₇	—	2	1	no	—
23	#127×C₉₇	C₂²	1	1	yes	characteristic
24	#123×C₉₇	—	2	2	no	—
25	C₂²⋊Q₈×C₉₇	C₂²	1	2	yes	—
26	#121×C₉₇	—	2	1	no	—
27	#121×C₉₇	—	2	2	no	—
28	#121×C₉₇	C₂²	1	2	yes	—
29	C₄×Q₈×C₉₇	C₂²	1	1	yes	characteristic
30	C₄×D₄×C₉₇	—	2	1	no	—
31	C₄×D₄×C₉₇	C₂²	1	1	yes	characteristic
32	C₄×D₄×C₉₇	C₂²	1	1	yes	characteristic
33	C₂×C₄⋊C₄×C₉₇	—	2	1	no	—
34	#106×C₉₇	C₂²	1	1	yes	characteristic
35	C₄⋊C₈×C₉₇	C₂²	1	1	yes	characteristic
36	#103×C₉₇	C₂²	1	1	yes	characteristic
37	#103×C₉₇	C₂²	1	2	yes	—
38	#102×C₉₇	C₂²	1	1	yes	characteristic
39	#102×C₉₇	C₂²	1	2	yes	—
40	C₂²⋊C₈×C₉₇	C₂²	1	2	yes	—
41	C₄×C₈×C₉₇	C₂²	1	1	yes	characteristic
42	#41×C₉₇	—	2	4	no	—
43	#41×C₉₇	—	2	4	no	—
44	#41×C₉₇	—	4	2	no	—
45	#41×C₉₇	—	4	2	no	—
46	C₂×Q₈×C₉₇	C₂³	1	2	yes	—
47	C₂×Q₈×C₉₇	C₂³	1	1	yes	characteristic
48	C₂×D₄×C₉₇	—	2	2	no	—
49	C₂×D₄×C₉₇	—	2	2	no	—
50	C₂×D₄×C₉₇	—	2	2	no	—
51	C₂×D₄×C₉₇	—	2	1	no	—
52	C₂×D₄×C₉₇	—	2	1	no	—
53	C₂×D₄×C₉₇	D₄	1	1	yes	characteristic
54	C₂×D₄×C₉₇	C₂³	1	2	yes	—
55	C₂×D₄×C₉₇	C₂³	1	1	yes	characteristic
56	C₂²×C₄×C₉₇	—	2	1	no	—
57	C₂²×C₄×C₉₇	—	2	2	no	—
58	C₂²×C₄×C₉₇	—	2	2	no	—
59	C₂²×C₄×C₉₇	C₂³	1	2	yes	—
60	Q₁₆×C₉₇	—	2	4	no	—
61	SD₁₆×C₉₇	—	2	4	no	—
62	SD₁₆×C₉₇	—	4	2	no	—
63	SD₁₆×C₉₇	D₄	1	4	yes	complemented
64	D₈×C₉₇	—	2	4	no	—
65	C₂×C₈×C₉₇	—	2	4	no	—
66	C₂×C₈×C₉₇	C₂³	1	1	yes	characteristic
67	C₂×C₈×C₉₇	C₂³	1	2	yes	—
68	C₂×C₈×C₉₇	C₂³	1	1	yes	characteristic
69	C₄⋊C₄×C₉₇	—	2	1	no	—
70	C₄⋊C₄×C₉₇	—	2	2	no	—
71	C₄⋊C₄×C₉₇	—	2	1	no	—
72	C₄⋊C₄×C₉₇	—	2	1	no	—
73	C₄⋊C₄×C₉₇	D₄	1	1	yes	characteristic
74	C₄⋊C₄×C₉₇	C₂³	1	1	yes	characteristic
75	C₄⋊C₄×C₉₇	C₂³	1	1	yes	characteristic
76	C₂²⋊C₄×C₉₇	—	2	2	no	—
77	C₂²⋊C₄×C₉₇	—	2	2	no	—
78	C₂²⋊C₄×C₉₇	—	2	1	no	—
79	C₂²⋊C₄×C₉₇	D₄	1	2	yes	—
80	C₄²×C₉₇	—	2	1	no	—
81	C₄²×C₉₇	C₂³	1	1	yes	characteristic
82	C₂³×C₉₇	—	2	1	no	—
83	C₂³×C₉₇	—	2	2	no	—
84	C₂³×C₉₇	C₂×D₄	1	2	yes	—
85	Q₈×C₉₇	—	2	1	no	—
86	Q₈×C₉₇	—	2	2	no	—
87	Q₈×C₉₇	—	2	2	no	—
88	Q₈×C₉₇	C₂×D₄	1	2	yes	—
89	D₄×C₉₇	—	2	2	no	—
90	D₄×C₉₇	—	2	2	no	—
91	D₄×C₉₇	—	2	1	no	—
92	D₄×C₉₇	—	2	2	no	—
93	D₄×C₉₇	—	4	4	no	—
94	D₄×C₉₇	—	4	4	no	—
95	D₄×C₉₇	—	4	2	no	—
96	D₄×C₉₇	—	8	1	no	—
97	D₄×C₉₇	—	8	2	no	—
98	D₄×C₉₇	C₂×D₄	1	2	yes	—
99	C₂×C₄×C₉₇	—	2	1	no	—
100	C₂×C₄×C₉₇	—	2	2	no	—
101	C₂×C₄×C₉₇	—	2	2	no	—
102	C₂×C₄×C₉₇	—	2	1	no	—
103	C₂×C₄×C₉₇	—	2	1	no	—
104	C₂×C₄×C₉₇	—	2	2	no	—
105	C₂×C₄×C₉₇	—	4	4	no	—
106	C₂×C₄×C₉₇	—	4	4	no	—
107	C₂×C₄×C₉₇	—	4	2	no	—
108	C₂×C₄×C₉₇	C₂×D₄	1	1	yes	characteristic
109	C₂×C₄×C₉₇	C₂×D₄	1	2	yes	—
110	C₂×C₄×C₉₇	C₂×D₄	1	1	yes	characteristic
111	C₂×C₄×C₉₇	C₂⁴	1	1	yes	characteristic
112	C₈×C₉₇	—	2	2	no	—
113	C₈×C₉₇	—	2	1	no	—
114	C₈×C₉₇	C₂×D₄	1	2	yes	—
115	C₂²×C₉₇	—	2	2	no	—
116	C₂²×C₉₇	—	2	2	no	—
117	C₂²×C₉₇	—	2	2	no	—
118	C₂²×C₉₇	—	2	2	no	—
119	C₂²×C₉₇	—	4	1	no	—
120	C₂²×C₉₇	—	4	1	no	—
121	C₂²×C₉₇	—	4	2	no	—
122	C₂²×C₉₇	—	4	2	no	—
123	C₂²×C₉₇	—	4	2	no	—
124	C₂²×C₉₇	C₂²×D₄	1	1	yes	characteristic
125	C₄×C₉₇	—	2	2	no	—
126	C₄×C₉₇	—	2	2	no	—
127	C₄×C₉₇	—	2	1	no	—
128	C₄×C₉₇	—	4	2	no	—
129	C₄×C₉₇	—	4	1	no	—
130	C₄×C₉₇	—	4	1	no	—
131	C₄×C₉₇	#135	1	2	yes	complemented
132	C₄×C₉₇	C₂²×D₄	1	1	yes	characteristic
133	C₄×C₉₇	H(5,𝔽₂)	1	1	yes	characteristic
134	C₂×C₉₇	—	4	2	no	—
135	C₂×C₉₇	—	4	2	no	—
136	C₂×C₉₇	—	8	2	no	—
137	C₂×C₉₇	D₄²	1	1	yes	characteristic
138	C₂×C₉₇	C₂×#135	1	1	yes	characteristic
139	C₂×C₉₇	#577	1	1	yes	characteristic
140	#3284	C₉₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
141	C₉₇	#3284	1	1	yes	characteristic, direct product factor, Sylow
142	C₂×#135	C₂×C₉₇	1	2	yes	complemented
143	#550	C₂×C₉₇	1	1	yes	characteristic, complemented
144	#547	C₂×C₉₇	1	1	yes	characteristic, complemented
145	#492	C₂×C₉₇	1	1	yes	characteristic, complemented
146	#465	C₂×C₉₇	1	2	yes	complemented
147	#462	C₂×C₉₇	1	1	yes	characteristic, complemented
148	#459	C₂×C₉₇	1	1	yes	characteristic, complemented
149	#452	C₂×C₉₇	1	2	yes	complemented
150	#449	C₂×C₉₇	1	2	yes	complemented
151	C₄×SD₁₆	C₂×C₉₇	1	1	yes	characteristic, complemented
152	C₈×D₄	C₂×C₉₇	1	1	yes	characteristic, complemented
153	C₂×#41	C₂²×C₉₇	1	2	yes	—
154	C₂×#41	C₂²×C₉₇	1	2	yes	—
155	#135	—	2	8	no	—
156	C₂×Q₁₆	C₂²×C₉₇	1	2	yes	—
157	C₂×SD₁₆	C₂²×C₉₇	1	2	yes	—
158	C₂×SD₁₆	C₂²×C₉₇	1	2	yes	—
159	C₂×SD₁₆	C₂²×C₉₇	1	1	yes	characteristic
160	C₂×D₈	C₂²×C₉₇	1	2	yes	—
161	C₂²×C₈	C₂²×C₉₇	1	2	yes	—
162	C₄⋊Q₈	C₂²×C₉₇	1	1	yes	characteristic
163	#127	—	2	1	no	—
164	#127	C₂²×C₉₇	1	1	yes	characteristic
165	#123	—	2	2	no	—
166	C₂²⋊Q₈	C₂²×C₉₇	1	2	yes	—
167	#121	—	2	1	no	—
168	#121	—	2	2	no	—
169	#121	C₂²×C₉₇	1	2	yes	—
170	C₄×Q₈	C₂²×C₉₇	1	1	yes	characteristic
171	C₄×D₄	—	2	1	no	—
172	C₄×D₄	C₂²×C₉₇	1	1	yes	characteristic
173	C₄×D₄	C₂²×C₉₇	1	1	yes	characteristic
174	C₂×C₄⋊C₄	—	2	1	no	—
175	#106	C₂²×C₉₇	1	1	yes	characteristic
176	C₄⋊C₈	C₂²×C₉₇	1	1	yes	characteristic
177	#103	C₂²×C₉₇	1	1	yes	characteristic
178	#103	C₂²×C₉₇	1	2	yes	—
179	#102	C₂²×C₉₇	1	1	yes	characteristic
180	#102	C₂²×C₉₇	1	2	yes	—
181	C₂²⋊C₈	C₂²×C₉₇	1	2	yes	—
182	C₄×C₈	C₂²×C₉₇	1	1	yes	characteristic
183	#41	—	2	4	no	—
184	#41	—	2	4	no	—
185	#41	—	4	2	no	—
186	#41	—	4	2	no	—
187	C₂×Q₈	C₂³×C₉₇	1	2	yes	—
188	C₂×Q₈	C₂³×C₉₇	1	1	yes	characteristic
189	C₂×D₄	—	2	2	no	—
190	C₂×D₄	—	2	2	no	—
191	C₂×D₄	—	2	2	no	—
192	C₂×D₄	—	2	1	no	—
193	C₂×D₄	—	2	1	no	—
194	C₂×D₄	D₄×C₉₇	1	1	yes	characteristic
195	C₂×D₄	C₂³×C₉₇	1	2	yes	—
196	C₂×D₄	C₂³×C₉₇	1	1	yes	characteristic
197	C₂²×C₄	—	2	1	no	—
198	C₂²×C₄	—	2	2	no	—
199	C₂²×C₄	—	2	2	no	—
200	C₂²×C₄	C₂³×C₉₇	1	2	yes	—
201	Q₁₆	—	2	4	no	—
202	SD₁₆	—	2	4	no	—
203	SD₁₆	—	4	2	no	—
204	SD₁₆	D₄×C₉₇	1	4	yes	complemented
205	D₈	—	2	4	no	—
206	C₂×C₈	—	2	4	no	—
207	C₂×C₈	C₂³×C₉₇	1	1	yes	characteristic
208	C₂×C₈	C₂³×C₉₇	1	2	yes	—
209	C₂×C₈	C₂³×C₉₇	1	1	yes	characteristic
210	C₄⋊C₄	—	2	1	no	—
211	C₄⋊C₄	—	2	2	no	—
212	C₄⋊C₄	—	2	1	no	—
213	C₄⋊C₄	—	2	1	no	—
214	C₄⋊C₄	D₄×C₉₇	1	1	yes	characteristic
215	C₄⋊C₄	C₂³×C₉₇	1	1	yes	characteristic
216	C₄⋊C₄	C₂³×C₉₇	1	1	yes	characteristic
217	C₂²⋊C₄	—	2	2	no	—
218	C₂²⋊C₄	—	2	2	no	—
219	C₂²⋊C₄	—	2	1	no	—
220	C₂²⋊C₄	D₄×C₉₇	1	2	yes	—
221	C₄²	—	2	1	no	—
222	C₄²	C₂³×C₉₇	1	1	yes	characteristic
223	C₂³	—	2	1	no	—
224	C₂³	—	2	2	no	—
225	C₂³	C₂×D₄×C₉₇	1	2	yes	—
226	Q₈	—	2	1	no	—
227	Q₈	—	2	2	no	—
228	Q₈	—	2	2	no	—
229	Q₈	C₂×D₄×C₉₇	1	2	yes	—
230	D₄	—	2	2	no	—
231	D₄	—	2	2	no	—
232	D₄	—	2	1	no	—
233	D₄	—	2	2	no	—
234	D₄	—	4	4	no	—
235	D₄	—	4	4	no	—
236	D₄	—	4	2	no	—
237	D₄	—	8	1	no	—
238	D₄	—	8	2	no	—
239	D₄	C₂×D₄×C₉₇	1	2	yes	—
240	C₂×C₄	—	2	1	no	—
241	C₂×C₄	—	2	2	no	—
242	C₂×C₄	—	2	2	no	—
243	C₂×C₄	—	2	1	no	—
244	C₂×C₄	—	2	1	no	—
245	C₂×C₄	—	2	2	no	—
246	C₂×C₄	—	4	4	no	—
247	C₂×C₄	—	4	4	no	—
248	C₂×C₄	—	4	2	no	—
249	C₂×C₄	C₂×D₄×C₉₇	1	1	yes	characteristic
250	C₂×C₄	C₂×D₄×C₉₇	1	2	yes	—
251	C₂×C₄	C₂×D₄×C₉₇	1	1	yes	characteristic
252	C₂×C₄	C₂⁴×C₉₇	1	1	yes	characteristic
253	C₈	—	2	2	no	—
254	C₈	—	2	1	no	—
255	C₈	C₂×D₄×C₉₇	1	2	yes	—
256	C₂²	—	2	2	no	—
257	C₂²	—	2	2	no	—
258	C₂²	—	2	2	no	—
259	C₂²	—	2	2	no	—
260	C₂²	—	4	1	no	—
261	C₂²	—	4	1	no	—
262	C₂²	—	4	2	no	—
263	C₂²	—	4	2	no	—
264	C₂²	—	4	2	no	—
265	C₂²	C₂²×D₄×C₉₇	1	1	yes	characteristic
266	C₄	—	2	2	no	—
267	C₄	—	2	2	no	—
268	C₄	—	2	1	no	—
269	C₄	—	4	2	no	—
270	C₄	—	4	1	no	—
271	C₄	—	4	1	no	—
272	C₄	#135×C₉₇	1	2	yes	complemented
273	C₄	C₂²×D₄×C₉₇	1	1	yes	characteristic
274	C₄	H(5,𝔽₂)×C₉₇	1	1	yes	characteristic
275	C₂	—	4	2	no	—
276	C₂	—	4	2	no	—
277	C₂	—	8	2	no	—
278	C₂	D₄²×C₉₇	1	1	yes	characteristic
279	C₂	C₂×#135×C₉₇	1	1	yes	characteristic
280	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×D₄×C₉₇ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×#135×C₉₇	C₂	semidirect
2. #550×C₉₇	C₂	semidirect
3. #547×C₉₇	C₂	semidirect
4. #492×C₉₇	C₂	semidirect
5. #465×C₉₇	C₂	semidirect
6. #462×C₉₇	C₂	semidirect
7. #459×C₉₇	C₂	semidirect
8. #452×C₉₇	C₂	semidirect
9. #449×C₉₇	C₂	semidirect
10. C₄×SD₁₆×C₉₇	C₂	semidirect
11. C₈×D₄×C₉₇	C₂	semidirect
63. SD₁₆×C₉₇	D₄	semidirect
131. C₄×C₉₇	#135	semidirect
140. #3284	C₉₇	direct (N × H)
141. C₉₇	#3284	direct (N × H)
142. C₂×#135	C₂×C₉₇	semidirect
143. #550	C₂×C₉₇	semidirect
144. #547	C₂×C₉₇	semidirect
145. #492	C₂×C₉₇	semidirect
146. #465	C₂×C₉₇	semidirect
147. #462	C₂×C₉₇	semidirect
148. #459	C₂×C₉₇	semidirect
149. #452	C₂×C₉₇	semidirect
150. #449	C₂×C₉₇	semidirect
151. C₄×SD₁₆	C₂×C₉₇	semidirect
152. C₈×D₄	C₂×C₉₇	semidirect
204. SD₁₆	D₄×C₉₇	semidirect
272. C₄	#135×C₉₇	semidirect

Maximal subgroups

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

subgroup	index
C₂×#135×C₉₇	2
#550×C₉₇	2
#547×C₉₇	2
#492×C₉₇	2
#465×C₉₇	2
#462×C₉₇	2
#459×C₉₇	2
#452×C₉₇	2
#449×C₉₇	2
C₄×SD₁₆×C₉₇	2
C₈×D₄×C₉₇	2
#3284	97

Subgroups — C97×#3284 #27139242

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₉₇×#3284 #27139242