Subgroups — C₉₇×#3276 #27139234

Order 12416.

472 conjugacy classes of subgroups.

The subgroup lattice has 270 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#137×C₉₇	C₂	1	2	yes	maximal, complemented
2	D₄×Q₈×C₉₇	C₂	1	1	yes	maximal, characteristic, complemented
3	#547×C₉₇	C₂	1	1	yes	maximal, characteristic
4	#494×C₉₇	C₂	1	1	yes	maximal, characteristic, complemented
5	#470×C₉₇	C₂	1	2	yes	maximal, complemented
6	#462×C₉₇	C₂	1	1	yes	maximal, characteristic, complemented
7	#461×C₉₇	C₂	1	1	yes	maximal, characteristic, complemented
8	#452×C₉₇	C₂	1	2	yes	maximal, complemented
9	#451×C₉₇	C₂	1	2	yes	maximal, complemented
10	C₄×SD₁₆×C₉₇	C₂	1	1	yes	maximal, characteristic, complemented
11	#436×C₉₇	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#41×C₉₇	C₂²	1	2	yes	—
13	C₂²×Q₈×C₉₇	C₂²	1	2	yes	—
14	#137×C₉₇	—	2	8	no	—
15	C₂×Q₁₆×C₉₇	C₂²	1	2	yes	—
16	C₂×Q₁₆×C₉₇	C₂²	1	2	yes	—
17	C₂×Q₁₆×C₉₇	C₂²	1	2	yes	—
18	C₂×SD₁₆×C₉₇	C₂²	1	2	yes	—
19	C₂×SD₁₆×C₉₇	C₂²	1	1	yes	characteristic
20	C₂×M₁₆×C₉₇	C₂²	1	2	yes	—
21	C₄⋊Q₈×C₉₇	—	2	1	no	—
22	C₄⋊Q₈×C₉₇	C₂²	1	1	yes	characteristic
23	C₄⋊Q₈×C₉₇	C₂²	1	1	yes	characteristic
24	#123×C₉₇	—	2	2	no	—
25	C₂²⋊Q₈×C₉₇	—	2	2	no	—
26	C₂²⋊Q₈×C₉₇	C₂²	1	2	yes	—
27	C₂²⋊Q₈×C₉₇	C₂²	1	2	yes	—
28	#121×C₉₇	—	2	1	no	—
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	#103×C₉₇	C₂²	1	2	yes	—
39	#102×C₉₇	C₂²	1	1	yes	characteristic
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	—	4	2	no	—
44	C₂×Q₈×C₉₇	—	2	2	no	—
45	C₂×Q₈×C₉₇	—	2	4	no	—
46	C₂×Q₈×C₉₇	—	4	2	no	—
47	C₂×Q₈×C₉₇	C₂³	1	2	yes	—
48	C₂×Q₈×C₉₇	C₂³	1	2	yes	—
49	C₂×Q₈×C₉₇	C₂³	1	1	yes	characteristic
50	C₂×D₄×C₉₇	—	2	2	no	—
51	C₂×D₄×C₉₇	D₄	1	1	yes	characteristic
52	C₂×D₄×C₉₇	C₂³	1	1	yes	characteristic
53	C₂²×C₄×C₉₇	—	2	2	no	—
54	C₂²×C₄×C₉₇	—	2	1	no	—
55	C₂²×C₄×C₉₇	—	2	2	no	—
56	C₂²×C₄×C₉₇	C₂³	1	2	yes	—
57	Q₁₆×C₉₇	—	2	4	no	—
58	Q₁₆×C₉₇	—	2	4	no	—
59	Q₁₆×C₉₇	—	4	2	no	—
60	SD₁₆×C₉₇	—	2	4	no	—
61	SD₁₆×C₉₇	D₄	1	4	yes	complemented
62	M₁₆×C₉₇	—	2	4	no	—
63	C₂×C₈×C₉₇	C₂³	1	1	yes	characteristic
64	C₂×C₈×C₉₇	C₂³	1	2	yes	—
65	C₂×C₈×C₉₇	C₂³	1	1	yes	characteristic
66	C₄⋊C₄×C₉₇	—	2	1	no	—
67	C₄⋊C₄×C₉₇	—	2	2	no	—
68	C₄⋊C₄×C₉₇	—	2	1	no	—
69	C₄⋊C₄×C₉₇	—	2	1	no	—
70	C₄⋊C₄×C₉₇	—	2	1	no	—
71	C₄⋊C₄×C₉₇	—	2	2	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	C₂×D₄	1	2	yes	—
84	Q₈×C₉₇	—	2	1	no	—
85	Q₈×C₉₇	—	2	2	no	—
86	Q₈×C₉₇	—	2	2	no	—
87	Q₈×C₉₇	—	2	2	no	—
88	Q₈×C₉₇	—	2	2	no	—
89	Q₈×C₉₇	—	4	4	no	—
90	Q₈×C₉₇	C₂×D₄	1	2	yes	—
91	D₄×C₉₇	—	2	2	no	—
92	D₄×C₉₇	—	2	1	no	—
93	D₄×C₉₇	—	4	4	no	—
94	D₄×C₉₇	C₂×D₄	1	2	yes	—
95	C₂×C₄×C₉₇	—	2	1	no	—
96	C₂×C₄×C₉₇	—	2	2	no	—
97	C₂×C₄×C₉₇	—	2	2	no	—
98	C₂×C₄×C₉₇	—	2	1	no	—
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	—	4	4	no	—
103	C₂×C₄×C₉₇	—	4	4	no	—
104	C₂×C₄×C₉₇	—	4	2	no	—
105	C₂×C₄×C₉₇	C₂×D₄	1	1	yes	characteristic
106	C₂×C₄×C₉₇	C₂×D₄	1	2	yes	—
107	C₂×C₄×C₉₇	C₂×D₄	1	1	yes	characteristic
108	C₂×C₄×C₉₇	C₂⁴	1	1	yes	characteristic
109	C₈×C₉₇	—	2	2	no	—
110	C₈×C₉₇	—	2	1	no	—
111	C₈×C₉₇	C₂×D₄	1	2	yes	—
112	C₂²×C₉₇	—	2	2	no	—
113	C₂²×C₉₇	—	2	2	no	—
114	C₂²×C₉₇	—	2	2	no	—
115	C₂²×C₉₇	—	2	2	no	—
116	C₂²×C₉₇	—	4	1	no	—
117	C₂²×C₉₇	—	4	1	no	—
118	C₂²×C₉₇	C₂²×D₄	1	1	yes	characteristic
119	C₄×C₉₇	—	2	2	no	—
120	C₄×C₉₇	—	2	2	no	—
121	C₄×C₉₇	—	2	1	no	—
122	C₄×C₉₇	—	4	2	no	—
123	C₄×C₉₇	—	4	2	no	—
124	C₄×C₉₇	—	4	1	no	—
125	C₄×C₉₇	—	4	1	no	—
126	C₄×C₉₇	#137	1	2	yes	complemented
127	C₄×C₉₇	C₂²×D₄	1	1	yes	characteristic
128	C₄×C₉₇	H(5,𝔽₂)	1	1	yes	characteristic
129	C₂×C₉₇	—	4	2	no	—
130	C₂×C₉₇	—	4	2	no	—
131	C₂×C₉₇	D₄²	1	1	yes	characteristic
132	C₂×C₉₇	C₂×#137	1	1	yes	characteristic
133	C₂×C₉₇	#578	1	1	yes	characteristic
134	#3276	C₉₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
135	C₉₇	#3276	1	1	yes	characteristic, direct product factor, Sylow
136	C₂×#137	C₂×C₉₇	1	2	yes	complemented
137	D₄×Q₈	C₂×C₉₇	1	1	yes	characteristic, complemented
138	#547	C₂×C₉₇	1	1	yes	characteristic
139	#494	C₂×C₉₇	1	1	yes	characteristic, complemented
140	#470	C₂×C₉₇	1	2	yes	complemented
141	#462	C₂×C₉₇	1	1	yes	characteristic, complemented
142	#461	C₂×C₉₇	1	1	yes	characteristic, complemented
143	#452	C₂×C₉₇	1	2	yes	complemented
144	#451	C₂×C₉₇	1	2	yes	complemented
145	C₄×SD₁₆	C₂×C₉₇	1	1	yes	characteristic, complemented
146	#436	C₂×C₉₇	1	1	yes	characteristic, complemented
147	C₂×#41	C₂²×C₉₇	1	2	yes	—
148	C₂²×Q₈	C₂²×C₉₇	1	2	yes	—
149	#137	—	2	8	no	—
150	C₂×Q₁₆	C₂²×C₉₇	1	2	yes	—
151	C₂×Q₁₆	C₂²×C₉₇	1	2	yes	—
152	C₂×Q₁₆	C₂²×C₉₇	1	2	yes	—
153	C₂×SD₁₆	C₂²×C₉₇	1	2	yes	—
154	C₂×SD₁₆	C₂²×C₉₇	1	1	yes	characteristic
155	C₂×M₁₆	C₂²×C₉₇	1	2	yes	—
156	C₄⋊Q₈	—	2	1	no	—
157	C₄⋊Q₈	C₂²×C₉₇	1	1	yes	characteristic
158	C₄⋊Q₈	C₂²×C₉₇	1	1	yes	characteristic
159	#123	—	2	2	no	—
160	C₂²⋊Q₈	—	2	2	no	—
161	C₂²⋊Q₈	C₂²×C₉₇	1	2	yes	—
162	C₂²⋊Q₈	C₂²×C₉₇	1	2	yes	—
163	#121	—	2	1	no	—
164	C₄×Q₈	C₂²×C₉₇	1	1	yes	characteristic
165	C₄×D₄	—	2	1	no	—
166	C₄×D₄	C₂²×C₉₇	1	1	yes	characteristic
167	C₄×D₄	C₂²×C₉₇	1	1	yes	characteristic
168	C₂×C₄⋊C₄	—	2	1	no	—
169	#106	C₂²×C₉₇	1	1	yes	characteristic
170	C₄⋊C₈	C₂²×C₉₇	1	1	yes	characteristic
171	#103	C₂²×C₉₇	1	1	yes	characteristic
172	#103	C₂²×C₉₇	1	2	yes	—
173	#103	C₂²×C₉₇	1	2	yes	—
174	#102	C₂²×C₉₇	1	1	yes	characteristic
175	C₂²⋊C₈	C₂²×C₉₇	1	2	yes	—
176	C₄×C₈	C₂²×C₉₇	1	1	yes	characteristic
177	#41	—	2	4	no	—
178	#41	—	4	2	no	—
179	C₂×Q₈	—	2	2	no	—
180	C₂×Q₈	—	2	4	no	—
181	C₂×Q₈	—	4	2	no	—
182	C₂×Q₈	C₂³×C₉₇	1	2	yes	—
183	C₂×Q₈	C₂³×C₉₇	1	2	yes	—
184	C₂×Q₈	C₂³×C₉₇	1	1	yes	characteristic
185	C₂×D₄	—	2	2	no	—
186	C₂×D₄	D₄×C₉₇	1	1	yes	characteristic
187	C₂×D₄	C₂³×C₉₇	1	1	yes	characteristic
188	C₂²×C₄	—	2	2	no	—
189	C₂²×C₄	—	2	1	no	—
190	C₂²×C₄	—	2	2	no	—
191	C₂²×C₄	C₂³×C₉₇	1	2	yes	—
192	Q₁₆	—	2	4	no	—
193	Q₁₆	—	2	4	no	—
194	Q₁₆	—	4	2	no	—
195	SD₁₆	—	2	4	no	—
196	SD₁₆	D₄×C₉₇	1	4	yes	complemented
197	M₁₆	—	2	4	no	—
198	C₂×C₈	C₂³×C₉₇	1	1	yes	characteristic
199	C₂×C₈	C₂³×C₉₇	1	2	yes	—
200	C₂×C₈	C₂³×C₉₇	1	1	yes	characteristic
201	C₄⋊C₄	—	2	1	no	—
202	C₄⋊C₄	—	2	2	no	—
203	C₄⋊C₄	—	2	1	no	—
204	C₄⋊C₄	—	2	1	no	—
205	C₄⋊C₄	—	2	1	no	—
206	C₄⋊C₄	—	2	2	no	—
207	C₄⋊C₄	—	2	1	no	—
208	C₄⋊C₄	D₄×C₉₇	1	1	yes	characteristic
209	C₄⋊C₄	C₂³×C₉₇	1	1	yes	characteristic
210	C₄⋊C₄	C₂³×C₉₇	1	1	yes	characteristic
211	C₂²⋊C₄	—	2	2	no	—
212	C₂²⋊C₄	—	2	2	no	—
213	C₂²⋊C₄	—	2	1	no	—
214	C₂²⋊C₄	D₄×C₉₇	1	2	yes	—
215	C₄²	—	2	1	no	—
216	C₄²	C₂³×C₉₇	1	1	yes	characteristic
217	C₂³	—	2	1	no	—
218	C₂³	C₂×D₄×C₉₇	1	2	yes	—
219	Q₈	—	2	1	no	—
220	Q₈	—	2	2	no	—
221	Q₈	—	2	2	no	—
222	Q₈	—	2	2	no	—
223	Q₈	—	2	2	no	—
224	Q₈	—	4	4	no	—
225	Q₈	C₂×D₄×C₉₇	1	2	yes	—
226	D₄	—	2	2	no	—
227	D₄	—	2	1	no	—
228	D₄	—	4	4	no	—
229	D₄	C₂×D₄×C₉₇	1	2	yes	—
230	C₂×C₄	—	2	1	no	—
231	C₂×C₄	—	2	2	no	—
232	C₂×C₄	—	2	2	no	—
233	C₂×C₄	—	2	1	no	—
234	C₂×C₄	—	2	1	no	—
235	C₂×C₄	—	2	2	no	—
236	C₂×C₄	—	2	2	no	—
237	C₂×C₄	—	4	4	no	—
238	C₂×C₄	—	4	4	no	—
239	C₂×C₄	—	4	2	no	—
240	C₂×C₄	C₂×D₄×C₉₇	1	1	yes	characteristic
241	C₂×C₄	C₂×D₄×C₉₇	1	2	yes	—
242	C₂×C₄	C₂×D₄×C₉₇	1	1	yes	characteristic
243	C₂×C₄	C₂⁴×C₉₇	1	1	yes	characteristic
244	C₈	—	2	2	no	—
245	C₈	—	2	1	no	—
246	C₈	C₂×D₄×C₉₇	1	2	yes	—
247	C₂²	—	2	2	no	—
248	C₂²	—	2	2	no	—
249	C₂²	—	2	2	no	—
250	C₂²	—	2	2	no	—
251	C₂²	—	4	1	no	—
252	C₂²	—	4	1	no	—
253	C₂²	C₂²×D₄×C₉₇	1	1	yes	characteristic
254	C₄	—	2	2	no	—
255	C₄	—	2	2	no	—
256	C₄	—	2	1	no	—
257	C₄	—	4	2	no	—
258	C₄	—	4	2	no	—
259	C₄	—	4	1	no	—
260	C₄	—	4	1	no	—
261	C₄	#137×C₉₇	1	2	yes	complemented
262	C₄	C₂²×D₄×C₉₇	1	1	yes	characteristic
263	C₄	H(5,𝔽₂)×C₉₇	1	1	yes	characteristic
264	C₂	—	4	2	no	—
265	C₂	—	4	2	no	—
266	C₂	D₄²×C₉₇	1	1	yes	characteristic
267	C₂	C₂×#137×C₉₇	1	1	yes	characteristic
268	C₂	#578×C₉₇	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₉₇	C₂²×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁴×C₉₇
prime radical	#547×C₉₇	C₂
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×#137×C₉₇	C₂	semidirect
2. D₄×Q₈×C₉₇	C₂	semidirect
4. #494×C₉₇	C₂	semidirect
5. #470×C₉₇	C₂	semidirect
6. #462×C₉₇	C₂	semidirect
7. #461×C₉₇	C₂	semidirect
8. #452×C₉₇	C₂	semidirect
9. #451×C₉₇	C₂	semidirect
10. C₄×SD₁₆×C₉₇	C₂	semidirect
11. #436×C₉₇	C₂	semidirect
61. SD₁₆×C₉₇	D₄	semidirect
126. C₄×C₉₇	#137	semidirect
134. #3276	C₉₇	direct (N × H)
135. C₉₇	#3276	direct (N × H)
136. C₂×#137	C₂×C₉₇	semidirect
137. D₄×Q₈	C₂×C₉₇	semidirect
139. #494	C₂×C₉₇	semidirect
140. #470	C₂×C₉₇	semidirect
141. #462	C₂×C₉₇	semidirect
142. #461	C₂×C₉₇	semidirect
143. #452	C₂×C₉₇	semidirect
144. #451	C₂×C₉₇	semidirect
145. C₄×SD₁₆	C₂×C₉₇	semidirect
146. #436	C₂×C₉₇	semidirect
196. SD₁₆	D₄×C₉₇	semidirect
261. C₄	#137×C₉₇	semidirect

Maximal subgroups

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

subgroup	index
C₂×#137×C₉₇	2
D₄×Q₈×C₉₇	2
#547×C₉₇	2
#494×C₉₇	2
#470×C₉₇	2
#462×C₉₇	2
#461×C₉₇	2
#452×C₉₇	2
#451×C₉₇	2
C₄×SD₁₆×C₉₇	2
#436×C₉₇	2
#3276	97

Subgroups — C97×#3276 #27139234

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₉₇×#3276 #27139234