Subgroups — C₉₇×#2952 #27138910

Order 12416.

484 conjugacy classes of subgroups.

The subgroup lattice has 182 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	1	yes	maximal, characteristic
2	#520×C₉₇	C₂	1	2	yes	maximal, complemented
3	#442×C₉₇	C₂	1	2	yes	maximal, complemented
4	#441×C₉₇	C₂	1	2	yes	maximal, complemented
5	C₄×SD₁₆×C₉₇	C₂	1	4	yes	maximal, complemented
6	C₂×#106×C₉₇	C₂	1	1	yes	maximal, characteristic, complemented
7	#417×C₉₇	C₂	1	2	yes	maximal, complemented
8	#405×C₉₇	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×#41×C₉₇	C₂²	1	2	yes	—
10	#135×C₉₇	C₄	1	8	yes	complemented
11	C₂×Q₁₆×C₉₇	C₂²	1	1	yes	characteristic
12	C₂×SD₁₆×C₉₇	C₂²	1	2	yes	—
13	C₂×D₈×C₉₇	C₂²	1	1	yes	characteristic
14	C₂×M₁₆×C₉₇	C₂²	1	1	yes	characteristic
15	C₂²×C₈×C₉₇	C₂²	1	1	yes	characteristic
16	C₄×Q₈×C₉₇	C₂²	1	4	yes	—
17	C₄×D₄×C₉₇	—	2	4	no	—
18	C₄×D₄×C₉₇	C₂²	1	4	yes	—
19	#117×C₉₇	—	2	2	no	—
20	#117×C₉₇	C₂²	1	1	yes	characteristic
21	C₂×C₄⋊C₄×C₉₇	—	2	2	no	—
22	C₂×C₄⋊C₄×C₉₇	C₂²	1	2	yes	—
23	#106×C₉₇	C₂²	1	2	yes	—
24	#106×C₉₇	C₂²	1	2	yes	—
25	#103×C₉₇	C₂²	1	4	yes	—
26	#102×C₉₇	C₂²	1	4	yes	—
27	#97×C₉₇	C₂²	1	2	yes	—
28	C₄×C₈×C₉₇	C₂²	1	2	yes	—
29	#41×C₉₇	—	2	4	no	—
30	#41×C₉₇	C₂×C₄	1	8	yes	complemented
31	C₂×Q₈×C₉₇	C₂³	1	2	yes	—
32	C₂×D₄×C₉₇	—	2	2	no	—
33	C₂×D₄×C₉₇	C₂³	1	2	yes	—
34	C₂²×C₄×C₉₇	—	2	2	no	—
35	C₂²×C₄×C₉₇	—	2	2	no	—
36	C₂²×C₄×C₉₇	—	2	4	no	—
37	C₂²×C₄×C₉₇	C₂³	1	1	yes	characteristic
38	Q₁₆×C₉₇	C₂×C₄	1	4	yes	complemented
39	SD₁₆×C₉₇	C₂×C₄	1	8	yes	complemented
40	D₈×C₉₇	C₂×C₄	1	4	yes	complemented
41	M₁₆×C₉₇	—	2	2	no	—
42	C₂×C₈×C₉₇	C₂×C₄	1	4	yes	complemented
43	C₂×C₈×C₉₇	C₂³	1	2	yes	—
44	C₂×C₈×C₉₇	C₂³	1	1	yes	characteristic
45	C₂×C₈×C₉₇	C₂³	1	1	yes	characteristic
46	C₄⋊C₄×C₉₇	—	2	2	no	—
47	C₄⋊C₄×C₉₇	—	2	4	no	—
48	C₄⋊C₄×C₉₇	D₄	1	2	yes	—
49	C₄⋊C₄×C₉₇	C₂³	1	4	yes	—
50	C₂²⋊C₄×C₉₇	—	2	4	no	—
51	C₂²⋊C₄×C₉₇	D₄	1	2	yes	—
52	C₄²×C₉₇	—	2	4	no	—
53	C₄²×C₉₇	C₂³	1	2	yes	—
54	C₂³×C₉₇	—	2	2	no	—
55	C₂³×C₉₇	C₂×D₄	1	1	yes	characteristic
56	Q₈×C₉₇	—	2	2	no	—
57	Q₈×C₉₇	C₂²×C₄	1	4	yes	—
58	D₄×C₉₇	—	2	8	no	—
59	D₄×C₉₇	—	2	2	no	—
60	D₄×C₉₇	C₂²×C₄	1	4	yes	—
61	C₂×C₄×C₉₇	—	2	2	no	—
62	C₂×C₄×C₉₇	—	2	8	no	—
63	C₂×C₄×C₉₇	—	2	4	no	—
64	C₂×C₄×C₉₇	—	8	4	no	—
65	C₂×C₄×C₉₇	—	8	2	no	—
66	C₂×C₄×C₉₇	C₂²×C₄	1	2	yes	—
67	C₂×C₄×C₉₇	C₂×D₄	1	4	yes	—
68	C₂×C₄×C₉₇	C₂×D₄	1	1	yes	characteristic
69	C₂×C₄×C₉₇	#41	1	2	yes	—
70	C₂×C₄×C₉₇	C₂⁴	1	1	yes	characteristic
71	C₈×C₉₇	—	2	2	no	—
72	C₈×C₉₇	C₂²×C₄	1	2	yes	—
73	C₈×C₉₇	C₂²×C₄	1	2	yes	—
74	C₂²×C₉₇	—	2	2	no	—
75	C₂²×C₉₇	—	2	4	no	—
76	C₂²×C₉₇	—	4	2	no	—
77	C₂²×C₉₇	—	4	2	no	—
78	C₂²×C₉₇	C₄×D₄	1	2	yes	—
79	C₂²×C₉₇	C₂²×D₄	1	1	yes	characteristic
80	C₄×C₉₇	—	2	4	no	—
81	C₄×C₉₇	—	2	4	no	—
82	C₄×C₉₇	—	4	4	no	—
83	C₄×C₉₇	C₄×D₄	1	2	yes	—
84	C₄×C₉₇	C₂³×C₄	1	1	yes	characteristic
85	C₄×C₉₇	C₂×#41	1	1	yes	characteristic
86	C₂×C₉₇	—	2	2	no	—
87	C₂×C₉₇	—	4	4	no	—
88	C₂×C₉₇	C₂×C₄×D₄	1	1	yes	characteristic
89	C₂×C₉₇	#577	1	2	yes	—
90	#2952	C₉₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
91	C₉₇	#2952	1	1	yes	characteristic, direct product factor, Sylow
92	C₂×#135	C₂×C₉₇	1	1	yes	characteristic
93	#520	C₂×C₉₇	1	2	yes	complemented
94	#442	C₂×C₉₇	1	2	yes	complemented
95	#441	C₂×C₉₇	1	2	yes	complemented
96	C₄×SD₁₆	C₂×C₉₇	1	4	yes	complemented
97	C₂×#106	C₂×C₉₇	1	1	yes	characteristic, complemented
98	#417	C₂×C₉₇	1	2	yes	complemented
99	#405	C₂×C₉₇	1	1	yes	characteristic, complemented
100	C₂×#41	C₂²×C₉₇	1	2	yes	—
101	#135	C₄×C₉₇	1	8	yes	complemented
102	C₂×Q₁₆	C₂²×C₉₇	1	1	yes	characteristic
103	C₂×SD₁₆	C₂²×C₉₇	1	2	yes	—
104	C₂×D₈	C₂²×C₉₇	1	1	yes	characteristic
105	C₂×M₁₆	C₂²×C₉₇	1	1	yes	characteristic
106	C₂²×C₈	C₂²×C₉₇	1	1	yes	characteristic
107	C₄×Q₈	C₂²×C₉₇	1	4	yes	—
108	C₄×D₄	—	2	4	no	—
109	C₄×D₄	C₂²×C₉₇	1	4	yes	—
110	#117	—	2	2	no	—
111	#117	C₂²×C₉₇	1	1	yes	characteristic
112	C₂×C₄⋊C₄	—	2	2	no	—
113	C₂×C₄⋊C₄	C₂²×C₉₇	1	2	yes	—
114	#106	C₂²×C₉₇	1	2	yes	—
115	#106	C₂²×C₉₇	1	2	yes	—
116	#103	C₂²×C₉₇	1	4	yes	—
117	#102	C₂²×C₉₇	1	4	yes	—
118	#97	C₂²×C₉₇	1	2	yes	—
119	C₄×C₈	C₂²×C₉₇	1	2	yes	—
120	#41	—	2	4	no	—
121	#41	C₂×C₄×C₉₇	1	8	yes	complemented
122	C₂×Q₈	C₂³×C₉₇	1	2	yes	—
123	C₂×D₄	—	2	2	no	—
124	C₂×D₄	C₂³×C₉₇	1	2	yes	—
125	C₂²×C₄	—	2	2	no	—
126	C₂²×C₄	—	2	2	no	—
127	C₂²×C₄	—	2	4	no	—
128	C₂²×C₄	C₂³×C₉₇	1	1	yes	characteristic
129	Q₁₆	C₂×C₄×C₉₇	1	4	yes	complemented
130	SD₁₆	C₂×C₄×C₉₇	1	8	yes	complemented
131	D₈	C₂×C₄×C₉₇	1	4	yes	complemented
132	M₁₆	—	2	2	no	—
133	C₂×C₈	C₂×C₄×C₉₇	1	4	yes	complemented
134	C₂×C₈	C₂³×C₉₇	1	2	yes	—
135	C₂×C₈	C₂³×C₉₇	1	1	yes	characteristic
136	C₂×C₈	C₂³×C₉₇	1	1	yes	characteristic
137	C₄⋊C₄	—	2	2	no	—
138	C₄⋊C₄	—	2	4	no	—
139	C₄⋊C₄	D₄×C₉₇	1	2	yes	—
140	C₄⋊C₄	C₂³×C₉₇	1	4	yes	—
141	C₂²⋊C₄	—	2	4	no	—
142	C₂²⋊C₄	D₄×C₉₇	1	2	yes	—
143	C₄²	—	2	4	no	—
144	C₄²	C₂³×C₉₇	1	2	yes	—
145	C₂³	—	2	2	no	—
146	C₂³	C₂×D₄×C₉₇	1	1	yes	characteristic
147	Q₈	—	2	2	no	—
148	Q₈	C₂²×C₄×C₉₇	1	4	yes	—
149	D₄	—	2	8	no	—
150	D₄	—	2	2	no	—
151	D₄	C₂²×C₄×C₉₇	1	4	yes	—
152	C₂×C₄	—	2	2	no	—
153	C₂×C₄	—	2	8	no	—
154	C₂×C₄	—	2	4	no	—
155	C₂×C₄	—	8	4	no	—
156	C₂×C₄	—	8	2	no	—
157	C₂×C₄	C₂²×C₄×C₉₇	1	2	yes	—
158	C₂×C₄	C₂×D₄×C₉₇	1	4	yes	—
159	C₂×C₄	C₂×D₄×C₉₇	1	1	yes	characteristic
160	C₂×C₄	#41×C₉₇	1	2	yes	—
161	C₂×C₄	C₂⁴×C₉₇	1	1	yes	characteristic
162	C₈	—	2	2	no	—
163	C₈	C₂²×C₄×C₉₇	1	2	yes	—
164	C₈	C₂²×C₄×C₉₇	1	2	yes	—
165	C₂²	—	2	2	no	—
166	C₂²	—	2	4	no	—
167	C₂²	—	4	2	no	—
168	C₂²	—	4	2	no	—
169	C₂²	C₄×D₄×C₉₇	1	2	yes	—
170	C₂²	C₂²×D₄×C₉₇	1	1	yes	characteristic
171	C₄	—	2	4	no	—
172	C₄	—	2	4	no	—
173	C₄	—	4	4	no	—
174	C₄	C₄×D₄×C₉₇	1	2	yes	—
175	C₄	C₂³×C₄×C₉₇	1	1	yes	characteristic
176	C₄	C₂×#41×C₉₇	1	1	yes	characteristic
177	C₂	—	2	2	no	—
178	C₂	—	4	4	no	—
179	C₂	C₂×C₄×D₄×C₉₇	1	1	yes	characteristic
180	C₂	#577×C₉₇	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₉₇	C₂²×D₄
commutator	C₄	C₂³×C₄×C₉₇
Frattini, cosocle	C₂×C₄	C₂⁴×C₉₇
prime radical	C₂×#135×C₉₇	C₂
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₉₇ ◁ #117×C₉₇ ◁ G

Complements

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

N	H ≅ G/N	type
2. #520×C₉₇	C₂	semidirect
3. #442×C₉₇	C₂	semidirect
4. #441×C₉₇	C₂	semidirect
5. C₄×SD₁₆×C₉₇	C₂	semidirect
6. C₂×#106×C₉₇	C₂	semidirect
7. #417×C₉₇	C₂	semidirect
8. #405×C₉₇	C₂	semidirect
10. #135×C₉₇	C₄	semidirect
30. #41×C₉₇	C₂×C₄	semidirect
38. Q₁₆×C₉₇	C₂×C₄	semidirect
39. SD₁₆×C₉₇	C₂×C₄	semidirect
40. D₈×C₉₇	C₂×C₄	semidirect
42. C₂×C₈×C₉₇	C₂×C₄	semidirect
90. #2952	C₉₇	direct (N × H)
91. C₉₇	#2952	direct (N × H)
93. #520	C₂×C₉₇	semidirect
94. #442	C₂×C₉₇	semidirect
95. #441	C₂×C₉₇	semidirect
96. C₄×SD₁₆	C₂×C₉₇	semidirect
97. C₂×#106	C₂×C₉₇	semidirect
98. #417	C₂×C₉₇	semidirect
99. #405	C₂×C₉₇	semidirect
101. #135	C₄×C₉₇	semidirect
121. #41	C₂×C₄×C₉₇	semidirect
129. Q₁₆	C₂×C₄×C₉₇	semidirect
130. SD₁₆	C₂×C₄×C₉₇	semidirect
131. D₈	C₂×C₄×C₉₇	semidirect
133. C₂×C₈	C₂×C₄×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
#520×C₉₇	2
#442×C₉₇	2
#441×C₉₇	2
C₄×SD₁₆×C₉₇	2
C₂×#106×C₉₇	2
#417×C₉₇	2
#405×C₉₇	2
#2952	97

Subgroups — C97×#2952 #27138910

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₉₇×#2952 #27138910