Subgroups — #38751686

Order 25216.

380 conjugacy classes of subgroups.

The subgroup lattice has 246 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#27161616	C₂	1	1	yes	maximal, characteristic
2	Q₈×#13096802	C₂	1	1	yes	maximal, characteristic, complemented
3	#27161158	C₂	1	1	yes	maximal, characteristic, complemented
4	#27161152	C₂	1	2	yes	maximal, complemented
5	#27161144	C₂	1	1	yes	maximal, characteristic, complemented
6	#27161144	C₂	1	2	yes	maximal, complemented
7	#27161139	C₂	1	2	yes	maximal, complemented
8	#27161135	C₂	1	1	yes	maximal, characteristic, complemented
9	#27161036	C₂	1	2	yes	maximal, complemented
10	C₄×C₈×D₁₉₇	C₂	1	1	yes	maximal, characteristic, complemented
11	#446×C₁₉₇	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#15691341	C₂²	1	2	yes	—
13	C₂×C₈×D₁₉₇	C₂²	1	1	yes	characteristic
14	C₂×C₈×D₁₉₇	C₂²	1	2	yes	—
15	C₂×C₈×D₁₉₇	C₂²	1	1	yes	characteristic
16	#20263321	C₄	1	2	yes	—
17	#20263317	C₄	1	1	yes	characteristic
18	#20263315	C₄	1	1	yes	characteristic
19	#20263311	C₄	1	2	yes	—
20	#20263302	C₄	1	1	yes	characteristic
21	#20263301	C₄	1	1	yes	characteristic
22	C₄×D₇₈₈	C₂²	1	2	yes	—
23	C₄×D₇₈₈	C₂²	1	1	yes	characteristic
24	#20263283	C₂²	1	2	yes	—
25	C₄²×D₁₉₇	C₂²	1	1	yes	characteristic
26	C₄⋊#13096802	C₂²	1	1	yes	characteristic
27	C₄⋊#13096802	C₂²	1	2	yes	—
28	#20263238	C₂²	1	2	yes	—
29	#20263238	C₂²	1	2	yes	—
30	#20263238	C₂²	1	2	yes	—
31	#20263236	C₂²	1	2	yes	—
32	C₄×#13096802	C₂²	1	1	yes	characteristic
33	C₄×#13096802	C₂²	1	2	yes	—
34	C₈×Dic₁₉₇	C₂²	1	1	yes	characteristic
35	C₈×Dic₁₉₇	C₂²	1	1	yes	characteristic
36	C₄×Q₈×C₁₉₇	C₂²	1	1	yes	characteristic
37	C₄⋊C₈×C₁₉₇	C₂²	1	1	yes	characteristic
38	C₄⋊C₈×C₁₉₇	C₂²	1	2	yes	—
39	#97×C₁₉₇	C₂²	1	2	yes	—
40	C₄×C₈×C₁₉₇	C₂²	1	1	yes	characteristic
41	C₂×D₇₈₈	C₂×C₄	1	2	yes	—
42	C₂×D₇₈₈	C₂×C₄	1	1	yes	characteristic
43	C₂×C₄×D₁₉₇	C₂×C₄	1	1	yes	characteristic
44	C₂×C₄×D₁₉₇	C₂×C₄	1	1	yes	characteristic
45	C₂×C₄×D₁₉₇	C₂³	1	1	yes	characteristic
46	C₂×C₄×D₁₉₇	C₂³	1	2	yes	—
47	#15691341	—	2	4	no	—
48	C₂×#13096802	C₂³	1	1	yes	characteristic
49	C₂×#13096802	C₂³	1	2	yes	—
50	C₂×#13096802	C₂³	1	1	yes	characteristic
51	C₈×D₁₉₇	—	2	2	no	—
52	C₈×D₁₉₇	—	2	4	no	—
53	C₈×D₁₉₇	—	2	2	no	—
54	C₄⋊Dic₁₉₇	C₂×C₄	1	1	yes	characteristic
55	C₄⋊Dic₁₉₇	C₂×C₄	1	2	yes	—
56	#15691337	C₂×C₄	1	2	yes	—
57	#15691337	C₂×C₄	1	2	yes	—
58	#15691335	C₂×C₄	1	2	yes	—
59	#15691335	C₂×C₄	1	2	yes	—
60	#15691335	C₂×C₄	1	2	yes	—
61	#15691335	C₂×C₄	1	2	yes	—
62	#15691335	C₂×C₄	1	2	yes	—
63	C₄×Dic₁₉₇	C₂×C₄	1	1	yes	characteristic
64	C₄×Dic₁₉₇	C₂×C₄	1	1	yes	characteristic
65	C₄×Dic₁₉₇	C₂³	1	1	yes	characteristic
66	C₂×Q₈×C₁₉₇	C₂×C₄	1	1	yes	characteristic
67	C₂×C₈×C₁₉₇	C₂³	1	1	yes	characteristic
68	C₂×C₈×C₁₉₇	C₂³	1	2	yes	—
69	C₂×C₈×C₁₉₇	C₂³	1	1	yes	characteristic
70	C₄⋊C₄×C₁₉₇	C₂×C₄	1	1	yes	characteristic
71	C₄⋊C₄×C₁₉₇	C₂×C₄	1	2	yes	—
72	C₄²×C₁₉₇	C₂³	1	1	yes	characteristic
73	C₄²×C₁₉₇	C₂³	1	2	yes	—
74	C₂²×D₁₉₇	C₂²×C₄	1	2	yes	—
75	C₂²×D₁₉₇	C₂²×C₄	1	1	yes	characteristic
76	D₇₈₈	—	2	4	no	—
77	D₇₈₈	—	2	2	no	—
78	C₂×Dic₁₉₇	C₂²×C₄	1	1	yes	characteristic
79	C₂×Dic₁₉₇	C₂²×C₄	1	1	yes	characteristic
80	C₂×Dic₁₉₇	C₂²×C₄	1	1	yes	characteristic
81	C₂×Dic₁₉₇	C₂²×C₄	1	2	yes	—
82	C₄×D₁₉₇	—	2	2	no	—
83	C₄×D₁₉₇	—	2	2	no	—
84	C₄×D₁₉₇	—	2	1	no	—
85	C₄×D₁₉₇	—	2	2	no	—
86	C₄×D₁₉₇	—	2	1	no	—
87	C₄×D₁₉₇	—	2	2	no	—
88	#13096802	—	2	1	no	—
89	#13096802	—	2	2	no	—
90	#13096802	#41	1	2	yes	—
91	Q₈×C₁₉₇	—	2	2	no	—
92	C₂×C₄×C₁₉₇	C₂²×C₄	1	1	yes	characteristic
93	C₂×C₄×C₁₉₇	C₂²×C₄	1	2	yes	—
94	C₂×C₄×C₁₉₇	C₂²×C₄	1	2	yes	—
95	C₂×C₄×C₁₉₇	C₂²×C₄	1	1	yes	characteristic
96	C₂×C₄×C₁₉₇	C₂⁴	1	1	yes	characteristic
97	C₈×C₁₉₇	—	2	1	no	—
98	C₈×C₁₉₇	—	2	2	no	—
99	C₈×C₁₉₇	#41	1	2	yes	—
100	C₂²×C₁₉₇	C₂³×C₄	1	1	yes	characteristic
101	C₂×D₁₉₇	—	2	1	no	—
102	C₂×D₁₉₇	—	2	1	no	—
103	C₂×D₁₉₇	—	2	2	no	—
104	C₂×D₁₉₇	—	2	2	no	—
105	C₂×D₁₉₇	—	2	2	no	—
106	C₂×D₁₉₇	#131	1	2	yes	—
107	C₄×C₁₉₇	—	2	2	no	—
108	C₄×C₁₉₇	—	2	1	no	—
109	C₄×C₁₉₇	—	2	2	no	—
110	C₄×C₁₉₇	#131	1	2	yes	—
111	C₄×C₁₉₇	C₂×#41	1	1	yes	characteristic
112	C₄×C₁₉₇	C₂×#41	1	1	yes	characteristic
113	Dic₁₉₇	—	2	1	no	—
114	Dic₁₉₇	—	2	2	no	—
115	Dic₁₉₇	—	2	1	no	—
116	Dic₁₉₇	—	2	1	no	—
117	C₂×C₁₉₇	C₄×#41	1	1	yes	characteristic
118	C₂×C₁₉₇	C₂×#131	1	1	yes	characteristic
119	C₂×C₁₉₇	C₂×#131	1	1	yes	characteristic
120	D₁₉₇	—	2	2	no	—
121	D₁₉₇	—	4	2	no	—
122	C₁₉₇	#2966	1	1	yes	characteristic, complemented, Sylow
123	#2966	—	197	1	no	maximal, Sylow
124	#529	—	197	1	no	—
125	#446	D₁₉₇	1	1	yes	characteristic, complemented
126	C₈×Q₈	—	197	1	no	—
127	#436	—	197	1	no	—
128	#435	—	197	2	no	—
129	C₈×D₄	—	197	1	no	—
130	C₈×D₄	—	197	2	no	—
131	#433	—	197	2	no	—
132	#432	—	197	1	no	—
133	#405	—	197	2	no	—
134	C₂×C₄×C₈	—	197	1	no	—
135	C₂×M₁₆	—	197	2	no	—
136	C₂²×C₈	—	197	1	no	—
137	C₂²×C₈	—	197	1	no	—
138	C₂²×C₈	—	197	2	no	—
139	#126	—	197	2	no	—
140	#125	—	197	1	no	—
141	#124	—	197	1	no	—
142	#123	—	197	2	no	—
143	C₂²⋊Q₈	—	197	1	no	—
144	#121	—	197	1	no	—
145	C₄×Q₈	C₂×D₁₉₇	1	1	yes	characteristic
146	C₄×D₄	—	197	2	no	—
147	C₄×D₄	—	197	1	no	—
148	#117	—	197	2	no	—
149	C₂×C₄²	—	197	1	no	—
150	C₄⋊C₈	—	197	1	no	—
151	C₄⋊C₈	—	197	2	no	—
152	C₄⋊C₈	C₂×D₁₉₇	1	1	yes	characteristic
153	C₄⋊C₈	C₂×D₁₉₇	1	2	yes	—
154	C₂²⋊C₈	—	197	2	no	—
155	C₂²⋊C₈	—	197	2	no	—
156	C₂²⋊C₈	—	197	2	no	—
157	#97	—	197	2	no	—
158	#97	C₂×D₁₉₇	1	2	yes	—
159	C₄×C₈	—	197	1	no	—
160	C₄×C₈	—	197	1	no	—
161	C₄×C₈	—	197	1	no	—
162	C₄×C₈	—	197	2	no	—
163	C₄×C₈	C₂×D₁₉₇	1	1	yes	characteristic
164	C₂×Q₈	C₄×D₁₉₇	1	1	yes	characteristic
165	C₂×D₄	—	197	2	no	—
166	C₂×D₄	—	197	1	no	—
167	C₂²×C₄	—	197	1	no	—
168	C₂²×C₄	—	197	1	no	—
169	C₂²×C₄	—	197	1	no	—
170	C₂²×C₄	—	197	2	no	—
171	M₁₆	—	394	4	no	—
172	C₂×C₈	—	394	2	no	—
173	C₂×C₈	—	394	4	no	—
174	C₂×C₈	—	394	2	no	—
175	C₂×C₈	—	197	1	no	—
176	C₂×C₈	—	197	2	no	—
177	C₂×C₈	—	197	1	no	—
178	C₂×C₈	C₂²×D₁₉₇	1	1	yes	characteristic
179	C₂×C₈	C₂²×D₁₉₇	1	2	yes	—
180	C₂×C₈	C₂²×D₁₉₇	1	1	yes	characteristic
181	C₄⋊C₄	—	197	2	no	—
182	C₄⋊C₄	—	197	2	no	—
183	C₄⋊C₄	—	197	1	no	—
184	C₄⋊C₄	—	197	2	no	—
185	C₄⋊C₄	C₄×D₁₉₇	1	1	yes	characteristic
186	C₄⋊C₄	C₄×D₁₉₇	1	2	yes	—
187	C₂²⋊C₄	—	197	2	no	—
188	C₂²⋊C₄	—	197	2	no	—
189	C₂²⋊C₄	—	197	2	no	—
190	C₂²⋊C₄	—	197	2	no	—
191	C₂²⋊C₄	—	197	2	no	—
192	C₄²	—	197	1	no	—
193	C₄²	—	197	1	no	—
194	C₄²	—	197	1	no	—
195	C₄²	C₂²×D₁₉₇	1	1	yes	characteristic
196	C₄²	C₂²×D₁₉₇	1	2	yes	—
197	C₂³	—	197	2	no	—
198	C₂³	—	197	1	no	—
199	Q₈	—	2	2	no	—
200	D₄	—	394	4	no	—
201	D₄	—	394	2	no	—
202	C₂×C₄	—	394	2	no	—
203	C₂×C₄	—	394	2	no	—
204	C₂×C₄	—	394	2	no	—
205	C₂×C₄	—	394	1	no	—
206	C₂×C₄	—	394	1	no	—
207	C₂×C₄	—	394	2	no	—
208	C₂×C₄	—	197	1	no	—
209	C₂×C₄	—	197	1	no	—
210	C₂×C₄	—	197	1	no	—
211	C₂×C₄	—	197	2	no	—
212	C₂×C₄	C₂×C₄×D₁₉₇	1	1	yes	characteristic
213	C₂×C₄	C₂×C₄×D₁₉₇	1	2	yes	—
214	C₂×C₄	C₂×C₄×D₁₉₇	1	2	yes	—
215	C₂×C₄	C₂×C₄×D₁₉₇	1	1	yes	characteristic
216	C₂×C₄	C₂³×D₁₉₇	1	1	yes	characteristic
217	C₈	—	2	1	no	—
218	C₈	—	2	2	no	—
219	C₈	—	394	1	no	—
220	C₈	—	394	2	no	—
221	C₈	—	197	2	no	—
222	C₈	#15691359	1	2	yes	—
223	C₂²	—	394	2	no	—
224	C₂²	—	394	2	no	—
225	C₂²	—	394	2	no	—
226	C₂²	—	394	1	no	—
227	C₂²	—	394	1	no	—
228	C₂²	—	197	2	no	—
229	C₂²	C₂²×C₄×D₁₉₇	1	1	yes	characteristic
230	C₄	—	2	1	no	—
231	C₄	—	2	2	no	—
232	C₄	—	2	2	no	—
233	C₄	—	394	1	no	—
234	C₄	—	394	2	no	—
235	C₄	—	394	1	no	—
236	C₄	—	394	1	no	—
237	C₄	#20263334	1	2	yes	—
238	C₄	#41×D₁₉₇	1	1	yes	characteristic
239	C₄	C₂×#15691359	1	1	yes	characteristic
240	C₂	—	788	2	no	—
241	C₂	—	394	2	no	—
242	C₂	C₄×#15691359	1	1	yes	characteristic
243	C₂	#131×D₁₉₇	1	1	yes	characteristic
244	C₂	C₂×#20263334	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₁₉₇
commutator, socle	C₂²×C₁₉₇	C₂³×C₄
Fitting	#446×C₁₉₇	C₂
prime radical	#20263301	C₄
cosocle	C₂×C₄×C₁₉₇	C₂⁴
hypercenter	#446	D₁₉₇
hypocenter	C₁₉₇	#2966
LCS	G ▷ C₂²×C₁₉₇ ▷ C₁₉₇
UCS	C₁ ◁ C₂×C₄ ◁ #446

Complements

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

N	H ≅ G/N	type
2. Q₈×#13096802	C₂	semidirect
3. #27161158	C₂	semidirect
4. #27161152	C₂	semidirect
5. #27161144	C₂	semidirect
6. #27161144	C₂	semidirect
7. #27161139	C₂	semidirect
8. #27161135	C₂	semidirect
9. #27161036	C₂	semidirect
10. C₄×C₈×D₁₉₇	C₂	semidirect
11. #446×C₁₉₇	C₂	semidirect
122. C₁₉₇	#2966	semidirect
125. #446	D₁₉₇	semidirect

Maximal subgroups

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

subgroup	index
#27161616	2
Q₈×#13096802	2
#27161158	2
#27161152	2
#27161144	2
#27161139	2
#27161135	2
#27161036	2
C₄×C₈×D₁₉₇	2
#446×C₁₉₇	2
#2966	197