Subgroups — #27144648

Order 12416.

568 conjugacy classes of subgroups.

The subgroup lattice has 200 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×Dic₁₉₄	C₂	1	1	yes	maximal, characteristic
2	C₂×#15690775	C₂	1	2	yes	maximal
3	#20260631	C₂	1	2	yes	maximal
4	#20260630	C₂	1	4	yes	maximal
5	#20260604	C₂	1	2	yes	maximal
6	C₄⋊C₄×Dic₉₇	C₂	1	2	yes	maximal
7	C₄×C₄⋊Dic₉₇	C₂	1	1	yes	maximal, characteristic
8	#384×C₉₇	C₂	1	1	yes	maximal, characteristic
9	C₂²×Dic₁₉₄	C₂²	1	1	yes	characteristic
10	C₄×Dic₁₉₄	C₄	1	8	yes	complemented
11	#15690775	—	2	8	no	—
12	C₂×C₄⋊Dic₉₇	C₂²	1	1	yes	characteristic
13	C₂×C₄⋊Dic₉₇	C₂²	1	1	yes	characteristic
14	C₂×C₄⋊Dic₉₇	C₂²	1	1	yes	characteristic
15	C₂×#13095490	C₂²	1	4	yes	—
16	C₂×#13095490	C₂²	1	4	yes	—
17	C₂×#13095490	C₂²	1	2	yes	—
18	C₂×C₄×Dic₉₇	C₂²	1	4	yes	—
19	C₂×C₄×Dic₉₇	C₂²	1	2	yes	—
20	C₂×C₄×Dic₉₇	C₂²	1	2	yes	—
21	#15690717	C₂²	1	2	yes	—
22	#15690717	C₂²	1	4	yes	—
23	C₂×C₄⋊C₄×C₉₇	C₂²	1	2	yes	—
24	C₂×C₄⋊C₄×C₉₇	C₂²	1	2	yes	—
25	C₂×C₄⋊C₄×C₉₇	C₂²	1	2	yes	—
26	C₂×C₄²×C₉₇	C₂²	1	1	yes	characteristic
27	C₂×Dic₁₉₄	C₂×C₄	1	4	yes	—
28	C₂×Dic₁₉₄	D₄	1	4	yes	—
29	C₂×Dic₁₉₄	Q₈	1	4	yes	—
30	C₂²×Dic₉₇	C₂³	1	2	yes	—
31	C₂²×Dic₉₇	C₂³	1	2	yes	—
32	C₂²×Dic₉₇	C₂³	1	2	yes	—
33	C₂²×Dic₉₇	C₂³	1	2	yes	—
34	C₄⋊Dic₉₇	—	2	2	no	—
35	C₄⋊Dic₉₇	C₂×C₄	1	4	yes	—
36	C₄⋊Dic₉₇	Q₈	1	4	yes	complemented
37	#13095490	—	2	8	no	—
38	#13095490	—	4	4	no	—
39	#13095490	C₂×C₄	1	8	yes	—
40	C₄×Dic₉₇	—	2	4	no	—
41	C₄×Dic₉₇	—	2	8	no	—
42	C₄×Dic₉₇	C₂×C₄	1	8	yes	—
43	C₂²×C₄×C₉₇	C₂³	1	4	yes	—
44	C₂²×C₄×C₉₇	C₂³	1	1	yes	characteristic
45	C₂²×C₄×C₉₇	C₂³	1	1	yes	characteristic
46	C₂²×C₄×C₉₇	C₂³	1	1	yes	characteristic
47	C₄⋊C₄×C₉₇	—	2	4	no	—
48	C₄⋊C₄×C₉₇	—	2	4	no	—
49	C₄⋊C₄×C₉₇	—	4	2	no	—
50	C₄²×C₉₇	C₂×C₄	1	4	yes	—
51	C₂³×C₉₇	C₂⁴	1	1	yes	characteristic
52	C₂×C₄×C₉₇	—	2	1	no	—
53	C₂×C₄×C₉₇	—	2	1	no	—
54	C₂×C₄×C₉₇	—	2	4	no	—
55	C₂×C₄×C₉₇	—	2	4	no	—
56	C₂×C₄×C₉₇	—	2	4	no	—
57	C₂×C₄×C₉₇	C₂²×C₄	1	2	yes	—
58	C₂×C₄×C₉₇	C₂²×C₄	1	2	yes	—
59	C₂×C₄×C₉₇	C₂²×C₄	1	2	yes	—
60	C₂×C₄×C₉₇	C₂×D₄	1	2	yes	—
61	C₂×C₄×C₉₇	C₂×Q₈	1	2	yes	—
62	C₂×C₄×C₉₇	C₂×Q₈	1	2	yes	—
63	C₂×C₄×C₉₇	#41	1	2	yes	—
64	C₂×Dic₉₇	—	2	2	no	—
65	C₂×Dic₉₇	—	2	2	no	—
66	C₂×Dic₉₇	—	2	2	no	—
67	C₂×Dic₉₇	—	2	2	no	—
68	C₂×Dic₉₇	—	2	2	no	—
69	C₂×Dic₉₇	—	2	2	no	—
70	C₂×Dic₉₇	—	2	2	no	—
71	C₂×Dic₉₇	C₂²×C₄	1	4	yes	—
72	C₂×Dic₉₇	C₂²×C₄	1	4	yes	—
73	C₂×Dic₉₇	C₂×D₄	1	4	yes	—
74	C₂×Dic₉₇	C₂×Q₈	1	4	yes	—
75	C₂×Dic₉₇	C₂×Q₈	1	4	yes	—
76	Dic₁₉₄	C₄⋊C₄	1	16	yes	complemented
77	C₂²×C₉₇	C₂³×C₄	1	1	yes	characteristic
78	C₂²×C₉₇	C₂²×D₄	1	1	yes	characteristic
79	C₂²×C₉₇	C₂²×Q₈	1	1	yes	characteristic
80	C₂²×C₉₇	C₂²×Q₈	1	1	yes	characteristic
81	C₂²×C₉₇	C₂×#41	1	1	yes	characteristic
82	C₂²×C₉₇	H(5,𝔽₂)	1	1	yes	characteristic
83	C₂²×C₉₇	2₋¹⁺⁴	1	1	yes	characteristic
84	C₄×C₉₇	—	2	2	no	—
85	C₄×C₉₇	—	4	4	no	—
86	C₄×C₉₇	C₂×C₄⋊C₄	1	4	yes	—
87	C₄×C₉₇	C₄×Q₈	1	4	yes	complemented
88	Dic₉₇	—	2	4	no	—
89	Dic₉₇	—	2	4	no	—
90	Dic₉₇	—	4	2	no	—
91	Dic₉₇	C₂×C₄⋊C₄	1	8	yes	—
92	C₂×C₉₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
93	C₂×C₉₇	C₂×C₄×Q₈	1	1	yes	characteristic
94	C₂×C₉₇	#520	1	1	yes	characteristic
95	C₂×C₉₇	D₄×Q₈	1	1	yes	characteristic
96	C₂×C₉₇	#550	1	1	yes	characteristic
97	C₂×C₉₇	#557	1	1	yes	characteristic
98	C₂×C₉₇	Q₈²	1	1	yes	characteristic
99	Q₈×C₄⋊C₄	—	97	1	no	maximal, Sylow
100	C₉₇	Q₈×C₄⋊C₄	1	1	yes	characteristic, complemented, Sylow
101	C₂×C₄×Q₈	—	97	2	no	—
102	C₂×C₄×Q₈	—	97	1	no	—
103	#389	—	97	4	no	—
104	#389	—	97	2	no	—
105	#384	—	97	2	no	—
106	#384	D₉₇	1	1	yes	characteristic
107	C₄×C₄⋊C₄	—	97	2	no	—
108	C₄×C₄⋊C₄	—	97	1	no	—
109	C₂²×Q₈	—	97	1	no	—
110	C₄×Q₈	—	97	8	no	—
111	C₄×Q₈	—	194	8	no	—
112	C₂×C₄⋊C₄	—	97	2	no	—
113	C₂×C₄⋊C₄	—	97	1	no	—
114	C₂×C₄⋊C₄	—	97	1	no	—
115	C₂×C₄⋊C₄	—	97	4	no	—
116	C₂×C₄⋊C₄	—	97	4	no	—
117	C₂×C₄⋊C₄	—	97	1	no	—
118	C₂×C₄⋊C₄	C₂×D₉₇	1	2	yes	—
119	C₂×C₄⋊C₄	C₂×D₉₇	1	2	yes	—
120	C₂×C₄⋊C₄	C₂×D₉₇	1	2	yes	—
121	C₂×C₄²	—	97	2	no	—
122	C₂×C₄²	—	97	4	no	—
123	C₂×C₄²	—	97	2	no	—
124	C₂×C₄²	C₂×D₉₇	1	1	yes	characteristic
125	#95	—	97	4	no	—
126	#95	—	97	2	no	—
127	C₂×Q₈	—	97	4	no	—
128	C₂×Q₈	—	97	4	no	—
129	C₂×Q₈	—	97	4	no	—
130	C₂²×C₄	—	97	2	no	—
131	C₂²×C₄	—	97	2	no	—
132	C₂²×C₄	—	97	2	no	—
133	C₂²×C₄	—	97	2	no	—
134	C₂²×C₄	C₂²×D₉₇	1	4	yes	—
135	C₂²×C₄	C₂²×D₉₇	1	1	yes	characteristic
136	C₂²×C₄	C₂²×D₉₇	1	1	yes	characteristic
137	C₂²×C₄	C₂²×D₉₇	1	1	yes	characteristic
138	C₄⋊C₄	—	2	4	no	—
139	C₄⋊C₄	—	2	4	no	—
140	C₄⋊C₄	—	4	2	no	—
141	C₄⋊C₄	—	97	8	no	—
142	C₄⋊C₄	—	97	4	no	—
143	C₄⋊C₄	—	97	4	no	—
144	C₄⋊C₄	—	388	4	no	—
145	C₄⋊C₄	—	194	8	no	—
146	C₄⋊C₄	—	194	2	no	—
147	C₄²	—	97	8	no	—
148	C₄²	—	194	8	no	—
149	C₄²	—	194	4	no	—
150	C₄²	C₄×D₉₇	1	4	yes	—
151	C₂³	C₂³×D₉₇	1	1	yes	characteristic
152	Q₈	—	97	16	no	—
153	C₂×C₄	—	2	1	no	—
154	C₂×C₄	—	2	1	no	—
155	C₂×C₄	—	2	4	no	—
156	C₂×C₄	—	2	4	no	—
157	C₂×C₄	—	2	4	no	—
158	C₂×C₄	—	97	4	no	—
159	C₂×C₄	—	97	4	no	—
160	C₂×C₄	—	97	4	no	—
161	C₂×C₄	—	97	4	no	—
162	C₂×C₄	—	97	4	no	—
163	C₂×C₄	—	194	2	no	—
164	C₂×C₄	—	194	2	no	—
165	C₂×C₄	—	194	2	no	—
166	C₂×C₄	—	194	2	no	—
167	C₂×C₄	—	194	2	no	—
168	C₂×C₄	—	194	2	no	—
169	C₂×C₄	—	194	2	no	—
170	C₂×C₄	C₂×C₄×D₉₇	1	2	yes	—
171	C₂×C₄	C₂×C₄×D₉₇	1	2	yes	—
172	C₂×C₄	C₂×C₄×D₉₇	1	2	yes	—
173	C₂×C₄	D₄×D₉₇	1	2	yes	—
174	C₂×C₄	Q₈×D₉₇	1	2	yes	—
175	C₂×C₄	Q₈×D₉₇	1	2	yes	—
176	C₂×C₄	#13095510	1	2	yes	—
177	C₂²	C₂²×C₄×D₉₇	1	1	yes	characteristic
178	C₂²	C₂×D₄×D₉₇	1	1	yes	characteristic
179	C₂²	C₂×Q₈×D₉₇	1	1	yes	characteristic
180	C₂²	C₂×Q₈×D₉₇	1	1	yes	characteristic
181	C₂²	C₂×#13095510	1	1	yes	characteristic
182	C₂²	#15690855	1	1	yes	characteristic
183	C₂²	#15690858	1	1	yes	characteristic
184	C₄	—	2	2	no	—
185	C₄	—	4	4	no	—
186	C₄	—	97	8	no	—
187	C₄	—	388	2	no	—
188	C₄	—	194	4	no	—
189	C₄	—	194	4	no	—
190	C₄	C₄⋊C₄×D₉₇	1	4	yes	—
191	C₄	#15690775	1	4	yes	complemented
192	C₂	C₂×C₄⋊C₄×D₉₇	1	1	yes	characteristic
193	C₂	C₂×#15690775	1	1	yes	characteristic
194	C₂	#20261216	1	1	yes	characteristic
195	C₂	#20261400	1	1	yes	characteristic
196	C₂	#20261405	1	1	yes	characteristic
197	C₂	#20261460	1	1	yes	characteristic
198	C₂	#20261465	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₉₇
commutator	C₂²×C₉₇	C₂³×C₄
Fitting	#384×C₉₇	C₂
socle, prime radical, cosocle	C₂³×C₉₇	C₂⁴
hypercenter	#384	D₉₇
hypocenter	C₉₇	Q₈×C₄⋊C₄
LCS	G ▷ C₂²×C₉₇ ▷ C₉₇
UCS	C₁ ◁ C₂³ ◁ #384

Complements

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

N	H ≅ G/N	type
10. C₄×Dic₁₉₄	C₄	semidirect
36. C₄⋊Dic₉₇	Q₈	semidirect
76. Dic₁₉₄	C₄⋊C₄	semidirect
87. C₄×C₉₇	C₄×Q₈	semidirect
100. C₉₇	Q₈×C₄⋊C₄	semidirect
191. C₄	#15690775	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄×Dic₁₉₄	2
C₂×#15690775	2
#20260631	2
#20260630	2
#20260604	2
C₄⋊C₄×Dic₉₇	2
C₄×C₄⋊Dic₉₇	2
#384×C₉₇	2
Q₈×C₄⋊C₄	97