Subgroups — C₉₇×#3095 #27139053

Order 12416.

410 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	C₂×C₂²⋊Q₈×C₉₇	C₂	1	1	yes	maximal, characteristic
2	C₂×#117×C₉₇	C₂	1	1	yes	maximal, characteristic
3	#485×C₉₇	C₂	1	4	yes	maximal, complemented
4	#484×C₉₇	C₂	1	2	yes	maximal, complemented
5	#483×C₉₇	C₂	1	2	yes	maximal, complemented
6	#428×C₉₇	C₂	1	2	yes	maximal, complemented
7	#419×C₉₇	C₂	1	2	yes	maximal, complemented
8	#407×C₉₇	C₂	1	1	yes	maximal, characteristic
9	C₂²×Q₈×C₉₇	C₂²	1	1	yes	characteristic
10	C₂³×C₄×C₉₇	C₂²	1	1	yes	characteristic
11	C₂×M₁₆×C₉₇	C₂²	1	2	yes	—
12	C₂²⋊Q₈×C₉₇	—	2	1	no	—
13	C₂²⋊Q₈×C₉₇	—	2	1	no	—
14	C₂²⋊Q₈×C₉₇	C₂²	1	2	yes	—
15	C₂²⋊Q₈×C₉₇	C₂²	1	2	yes	—
16	#117×C₉₇	—	2	1	no	—
17	#117×C₉₇	—	2	1	no	—
18	#117×C₉₇	C₂²	1	2	yes	—
19	#117×C₉₇	C₂²	1	2	yes	—
20	C₂×C₄⋊C₄×C₉₇	—	2	1	no	—
21	C₂×C₄⋊C₄×C₉₇	C₂²	1	1	yes	characteristic
22	C₂×C₄⋊C₄×C₉₇	C₂²	1	1	yes	characteristic
23	C₂×C₄⋊C₄×C₉₇	C₂²	1	1	yes	characteristic
24	C₂×C₂²⋊C₄×C₉₇	—	2	1	no	—
25	C₂×C₂²⋊C₄×C₉₇	—	2	1	no	—
26	C₂×C₄²×C₉₇	—	2	1	no	—
27	#107×C₉₇	C₂²	1	4	yes	complemented
28	#106×C₉₇	C₂²	1	4	yes	complemented
29	#103×C₉₇	C₂²	1	4	yes	complemented
30	#103×C₉₇	C₂²	1	4	yes	complemented
31	C₂²⋊C₈×C₉₇	C₂²	1	2	yes	—
32	C₂²⋊C₈×C₉₇	C₂²	1	2	yes	—
33	C₂⁴×C₉₇	D₄	1	1	yes	characteristic
34	C₂×Q₈×C₉₇	—	2	1	no	—
35	C₂×Q₈×C₉₇	—	2	4	no	—
36	C₂×Q₈×C₉₇	C₂³	1	2	yes	—
37	C₂²×C₄×C₉₇	—	2	4	no	—
38	C₂²×C₄×C₉₇	—	2	1	no	—
39	C₂²×C₄×C₉₇	—	2	1	no	—
40	C₂²×C₄×C₉₇	—	2	1	no	—
41	C₂²×C₄×C₉₇	—	2	1	no	—
42	C₂²×C₄×C₉₇	D₄	1	1	yes	characteristic
43	C₂²×C₄×C₉₇	D₄	1	1	yes	characteristic
44	C₂²×C₄×C₉₇	D₄	1	1	yes	characteristic
45	C₂²×C₄×C₉₇	C₂³	1	1	yes	characteristic
46	C₂²×C₄×C₉₇	C₂³	1	1	yes	characteristic
47	C₂²×C₄×C₉₇	C₂³	1	1	yes	characteristic
48	M₁₆×C₉₇	—	2	4	no	—
49	C₂×C₈×C₉₇	C₂³	1	4	yes	—
50	C₄⋊C₄×C₉₇	—	2	2	no	—
51	C₄⋊C₄×C₉₇	—	2	2	no	—
52	C₄⋊C₄×C₉₇	—	2	1	no	—
53	C₄⋊C₄×C₉₇	—	2	1	no	—
54	C₄⋊C₄×C₉₇	—	2	1	no	—
55	C₄⋊C₄×C₉₇	C₂³	1	2	yes	—
56	C₄⋊C₄×C₉₇	C₂³	1	2	yes	—
57	C₄⋊C₄×C₉₇	C₂³	1	2	yes	—
58	C₂²⋊C₄×C₉₇	—	2	2	no	—
59	C₂²⋊C₄×C₉₇	—	2	2	no	—
60	C₂²⋊C₄×C₉₇	—	2	2	no	—
61	C₂²⋊C₄×C₉₇	—	2	2	no	—
62	C₄²×C₉₇	—	2	2	no	—
63	C₄²×C₉₇	—	2	2	no	—
64	C₂³×C₉₇	—	2	2	no	—
65	C₂³×C₉₇	—	2	2	no	—
66	C₂³×C₉₇	—	4	1	no	—
67	C₂³×C₉₇	C₂×D₄	1	1	yes	characteristic
68	C₂³×C₉₇	C₂×D₄	1	1	yes	characteristic
69	C₂³×C₉₇	C₂×D₄	1	1	yes	characteristic
70	Q₈×C₉₇	—	2	4	no	—
71	Q₈×C₉₇	—	4	2	no	—
72	C₂×C₄×C₉₇	—	2	2	no	—
73	C₂×C₄×C₉₇	—	2	2	no	—
74	C₂×C₄×C₉₇	—	2	2	no	—
75	C₂×C₄×C₉₇	—	2	2	no	—
76	C₂×C₄×C₉₇	—	2	2	no	—
77	C₂×C₄×C₉₇	—	2	2	no	—
78	C₂×C₄×C₉₇	—	2	2	no	—
79	C₂×C₄×C₉₇	—	2	2	no	—
80	C₂×C₄×C₉₇	—	2	2	no	—
81	C₂×C₄×C₉₇	—	4	2	no	—
82	C₂×C₄×C₉₇	—	4	2	no	—
83	C₂×C₄×C₉₇	—	4	2	no	—
84	C₂×C₄×C₉₇	—	4	2	no	—
85	C₂×C₄×C₉₇	C₂×D₄	1	1	yes	characteristic
86	C₂×C₄×C₉₇	C₂×D₄	1	1	yes	characteristic
87	C₂×C₄×C₉₇	C₂×D₄	1	1	yes	characteristic
88	C₂×C₄×C₉₇	#41	1	4	yes	—
89	C₂×C₄×C₉₇	C₂⁴	1	1	yes	characteristic
90	C₈×C₉₇	—	2	4	no	—
91	C₂²×C₉₇	—	2	1	no	—
92	C₂²×C₉₇	—	2	1	no	—
93	C₂²×C₉₇	—	2	1	no	—
94	C₂²×C₉₇	—	2	1	no	—
95	C₂²×C₉₇	—	2	1	no	—
96	C₂²×C₉₇	—	2	1	no	—
97	C₂²×C₉₇	—	2	1	no	—
98	C₂²×C₉₇	—	4	4	no	—
99	C₂²×C₉₇	#123	1	2	yes	—
100	C₂²×C₉₇	#137	1	2	yes	—
101	C₂²×C₉₇	C₂²×D₄	1	1	yes	characteristic
102	C₄×C₉₇	—	2	1	no	—
103	C₄×C₉₇	—	2	1	no	—
104	C₄×C₉₇	—	4	2	no	—
105	C₄×C₉₇	—	4	2	no	—
106	C₄×C₉₇	—	4	2	no	—
107	C₄×C₉₇	—	4	2	no	—
108	C₄×C₉₇	#123	1	2	yes	—
109	C₄×C₉₇	C₂×#41	1	2	yes	—
110	C₂×C₉₇	—	2	2	no	—
111	C₂×C₉₇	—	2	2	no	—
112	C₂×C₉₇	—	4	1	no	—
113	C₂×C₉₇	C₂×#123	1	1	yes	characteristic
114	C₂×C₉₇	C₂×#137	1	1	yes	characteristic
115	C₂×C₉₇	#575	1	1	yes	characteristic
116	#3095	C₉₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
117	C₉₇	#3095	1	1	yes	characteristic, direct product factor, Sylow
118	C₂×C₂²⋊Q₈	C₂×C₉₇	1	1	yes	characteristic
119	C₂×#117	C₂×C₉₇	1	1	yes	characteristic
120	#485	C₂×C₉₇	1	4	yes	complemented
121	#484	C₂×C₉₇	1	2	yes	complemented
122	#483	C₂×C₉₇	1	2	yes	complemented
123	#428	C₂×C₉₇	1	2	yes	complemented
124	#419	C₂×C₉₇	1	2	yes	complemented
125	#407	C₂×C₉₇	1	1	yes	characteristic
126	C₂²×Q₈	C₂²×C₉₇	1	1	yes	characteristic
127	C₂³×C₄	C₂²×C₉₇	1	1	yes	characteristic
128	C₂×M₁₆	C₂²×C₉₇	1	2	yes	—
129	C₂²⋊Q₈	—	2	1	no	—
130	C₂²⋊Q₈	—	2	1	no	—
131	C₂²⋊Q₈	C₂²×C₉₇	1	2	yes	—
132	C₂²⋊Q₈	C₂²×C₉₇	1	2	yes	—
133	#117	—	2	1	no	—
134	#117	—	2	1	no	—
135	#117	C₂²×C₉₇	1	2	yes	—
136	#117	C₂²×C₉₇	1	2	yes	—
137	C₂×C₄⋊C₄	—	2	1	no	—
138	C₂×C₄⋊C₄	C₂²×C₉₇	1	1	yes	characteristic
139	C₂×C₄⋊C₄	C₂²×C₉₇	1	1	yes	characteristic
140	C₂×C₄⋊C₄	C₂²×C₉₇	1	1	yes	characteristic
141	C₂×C₂²⋊C₄	—	2	1	no	—
142	C₂×C₂²⋊C₄	—	2	1	no	—
143	C₂×C₄²	—	2	1	no	—
144	#107	C₂²×C₉₇	1	4	yes	complemented
145	#106	C₂²×C₉₇	1	4	yes	complemented
146	#103	C₂²×C₉₇	1	4	yes	complemented
147	#103	C₂²×C₉₇	1	4	yes	complemented
148	C₂²⋊C₈	C₂²×C₉₇	1	2	yes	—
149	C₂²⋊C₈	C₂²×C₉₇	1	2	yes	—
150	C₂⁴	D₄×C₉₇	1	1	yes	characteristic
151	C₂×Q₈	—	2	1	no	—
152	C₂×Q₈	—	2	4	no	—
153	C₂×Q₈	C₂³×C₉₇	1	2	yes	—
154	C₂²×C₄	—	2	4	no	—
155	C₂²×C₄	—	2	1	no	—
156	C₂²×C₄	—	2	1	no	—
157	C₂²×C₄	—	2	1	no	—
158	C₂²×C₄	—	2	1	no	—
159	C₂²×C₄	D₄×C₉₇	1	1	yes	characteristic
160	C₂²×C₄	D₄×C₉₇	1	1	yes	characteristic
161	C₂²×C₄	D₄×C₉₇	1	1	yes	characteristic
162	C₂²×C₄	C₂³×C₉₇	1	1	yes	characteristic
163	C₂²×C₄	C₂³×C₉₇	1	1	yes	characteristic
164	C₂²×C₄	C₂³×C₉₇	1	1	yes	characteristic
165	M₁₆	—	2	4	no	—
166	C₂×C₈	C₂³×C₉₇	1	4	yes	—
167	C₄⋊C₄	—	2	2	no	—
168	C₄⋊C₄	—	2	2	no	—
169	C₄⋊C₄	—	2	1	no	—
170	C₄⋊C₄	—	2	1	no	—
171	C₄⋊C₄	—	2	1	no	—
172	C₄⋊C₄	C₂³×C₉₇	1	2	yes	—
173	C₄⋊C₄	C₂³×C₉₇	1	2	yes	—
174	C₄⋊C₄	C₂³×C₉₇	1	2	yes	—
175	C₂²⋊C₄	—	2	2	no	—
176	C₂²⋊C₄	—	2	2	no	—
177	C₂²⋊C₄	—	2	2	no	—
178	C₂²⋊C₄	—	2	2	no	—
179	C₄²	—	2	2	no	—
180	C₄²	—	2	2	no	—
181	C₂³	—	2	2	no	—
182	C₂³	—	2	2	no	—
183	C₂³	—	4	1	no	—
184	C₂³	C₂×D₄×C₉₇	1	1	yes	characteristic
185	C₂³	C₂×D₄×C₉₇	1	1	yes	characteristic
186	C₂³	C₂×D₄×C₉₇	1	1	yes	characteristic
187	Q₈	—	2	4	no	—
188	Q₈	—	4	2	no	—
189	C₂×C₄	—	2	2	no	—
190	C₂×C₄	—	2	2	no	—
191	C₂×C₄	—	2	2	no	—
192	C₂×C₄	—	2	2	no	—
193	C₂×C₄	—	2	2	no	—
194	C₂×C₄	—	2	2	no	—
195	C₂×C₄	—	2	2	no	—
196	C₂×C₄	—	2	2	no	—
197	C₂×C₄	—	2	2	no	—
198	C₂×C₄	—	4	2	no	—
199	C₂×C₄	—	4	2	no	—
200	C₂×C₄	—	4	2	no	—
201	C₂×C₄	—	4	2	no	—
202	C₂×C₄	C₂×D₄×C₉₇	1	1	yes	characteristic
203	C₂×C₄	C₂×D₄×C₉₇	1	1	yes	characteristic
204	C₂×C₄	C₂×D₄×C₉₇	1	1	yes	characteristic
205	C₂×C₄	#41×C₉₇	1	4	yes	—
206	C₂×C₄	C₂⁴×C₉₇	1	1	yes	characteristic
207	C₈	—	2	4	no	—
208	C₂²	—	2	1	no	—
209	C₂²	—	2	1	no	—
210	C₂²	—	2	1	no	—
211	C₂²	—	2	1	no	—
212	C₂²	—	2	1	no	—
213	C₂²	—	2	1	no	—
214	C₂²	—	2	1	no	—
215	C₂²	—	4	4	no	—
216	C₂²	#123×C₉₇	1	2	yes	—
217	C₂²	#137×C₉₇	1	2	yes	—
218	C₂²	C₂²×D₄×C₉₇	1	1	yes	characteristic
219	C₄	—	2	1	no	—
220	C₄	—	2	1	no	—
221	C₄	—	4	2	no	—
222	C₄	—	4	2	no	—
223	C₄	—	4	2	no	—
224	C₄	—	4	2	no	—
225	C₄	#123×C₉₇	1	2	yes	—
226	C₄	C₂×#41×C₉₇	1	2	yes	—
227	C₂	—	2	2	no	—
228	C₂	—	2	2	no	—
229	C₂	—	4	1	no	—
230	C₂	C₂×#123×C₉₇	1	1	yes	characteristic
231	C₂	C₂×#137×C₉₇	1	1	yes	characteristic
232	C₂	#575×C₉₇	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₉₇	C₂²×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁴×C₉₇
prime radical	C₂⁴×C₉₇	D₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₉₇ ◁ C₂³×C₄×C₉₇ ◁ G

Complements

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

N	H ≅ G/N	type
3. #485×C₉₇	C₂	semidirect
4. #484×C₉₇	C₂	semidirect
5. #483×C₉₇	C₂	semidirect
6. #428×C₉₇	C₂	semidirect
7. #419×C₉₇	C₂	semidirect
27. #107×C₉₇	C₂²	semidirect
28. #106×C₉₇	C₂²	semidirect
29. #103×C₉₇	C₂²	semidirect
30. #103×C₉₇	C₂²	semidirect
116. #3095	C₉₇	direct (N × H)
117. C₉₇	#3095	direct (N × H)
120. #485	C₂×C₉₇	semidirect
121. #484	C₂×C₉₇	semidirect
122. #483	C₂×C₉₇	semidirect
123. #428	C₂×C₉₇	semidirect
124. #419	C₂×C₉₇	semidirect
144. #107	C₂²×C₉₇	semidirect
145. #106	C₂²×C₉₇	semidirect
146. #103	C₂²×C₉₇	semidirect
147. #103	C₂²×C₉₇	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₂²⋊Q₈×C₉₇	2
C₂×#117×C₉₇	2
#485×C₉₇	2
#484×C₉₇	2
#483×C₉₇	2
#428×C₉₇	2
#419×C₉₇	2
#407×C₉₇	2
#3095	97

Subgroups — C97×#3095 #27139053

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₉₇×#3095 #27139053