Subgroups — #41×Dic₂₇₈ #31819984

Order 17792.

1068 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	Q₈×Dic₂₇₈	C₂	1	1	yes	maximal, characteristic, complemented
2	#22581682	C₂	1	1	yes	maximal, characteristic, complemented
3	#22581679	C₂	1	2	yes	maximal, complemented
4	#22581662	C₂	1	6	yes	maximal, complemented
5	#22581660	C₂	1	3	yes	maximal, complemented
6	D₄×Dic₂₇₈	C₂	1	3	yes	maximal, complemented
7	#22581510	C₂	1	3	yes	maximal, complemented
8	#22581506	C₂	1	6	yes	maximal, complemented
9	#41×Dic₁₃₉	C₂	1	2	yes	maximal
10	C₂×C₄×Dic₂₇₈	C₂	1	3	yes	maximal, complemented
11	C₄×#41×C₁₃₉	C₂	1	1	yes	maximal, characteristic
12	C₂²×Dic₂₇₈	C₂²	1	3	yes	—
13	#17984842	C₂²	1	3	yes	—
14	#17984842	C₂²	1	3	yes	—
15	#17984841	C₂²	1	6	yes	—
16	#17984841	C₂²	1	6	yes	—
17	#17984834	C₂²	1	6	yes	—
18	#17984833	C₂²	1	6	yes	—
19	#17984833	C₂²	1	6	yes	—
20	#17984821	C₂²	1	6	yes	—
21	#17984821	C₂²	1	6	yes	—
22	#17984819	C₂²	1	12	yes	—
23	#17984819	C₂²	1	12	yes	—
24	Q₈×Dic₁₃₉	C₂²	1	2	yes	—
25	Q₈×Dic₁₃₉	C₂²	1	2	yes	—
26	C₄×Dic₂₇₈	C₂²	1	6	yes	—
27	C₄×Dic₂₇₈	C₂²	1	6	yes	—
28	C₄×Dic₂₇₈	C₂²	1	3	yes	—
29	C₄×Dic₂₇₈	C₂²	1	3	yes	—
30	C₄×Dic₂₇₈	C₂²	1	3	yes	—
31	C₄×Dic₂₇₈	C₂²	1	6	yes	—
32	C₄×Dic₂₇₈	C₂²	1	1	yes	characteristic
33	D₄×Dic₁₃₉	C₂²	1	6	yes	—
34	D₄×Dic₁₃₉	C₂²	1	6	yes	—
35	#17984801	C₂²	1	6	yes	—
36	C₂×C₄⋊Dic₁₃₉	C₂²	1	3	yes	—
37	C₂×Dic₁₃₉⋊C₄	C₂²	1	6	yes	—
38	C₂×C₄×Dic₁₃₉	C₂²	1	6	yes	—
39	C₂×#41×C₁₃₉	C₂²	1	1	yes	characteristic
40	C₄×Q₈×C₁₃₉	C₂²	1	1	yes	characteristic
41	C₄×Q₈×C₁₃₉	C₂²	1	1	yes	characteristic
42	C₄×D₄×C₁₃₉	C₂²	1	3	yes	—
43	C₄×D₄×C₁₃₉	C₂²	1	3	yes	—
44	#117×C₁₃₉	C₂²	1	3	yes	—
45	C₂×C₄²×C₁₃₉	C₂²	1	3	yes	—
46	C₂×Dic₂₇₈	—	2	12	no	—
47	C₂×Dic₂₇₈	C₂³	1	6	yes	—
48	C₂×Dic₂₇₈	C₂³	1	3	yes	—
49	C₂×Dic₂₇₈	C₂³	1	6	yes	—
50	C₂×Dic₂₇₈	C₂³	1	1	yes	characteristic
51	C₂²×Dic₁₃₉	C₂³	1	6	yes	—
52	C₂²×Dic₁₃₉	C₂³	1	6	yes	—
53	C₄⋊Dic₁₃₉	C₂³	1	1	yes	characteristic
54	C₄⋊Dic₁₃₉	C₂³	1	6	yes	—
55	C₄⋊Dic₁₃₉	C₂³	1	3	yes	—
56	C₄⋊Dic₁₃₉	C₂³	1	6	yes	—
57	C₄⋊Dic₁₃₉	C₂³	1	6	yes	—
58	C₄⋊Dic₁₃₉	C₂³	1	6	yes	—
59	Dic₁₃₉⋊C₄	C₂³	1	2	yes	—
60	Dic₁₃₉⋊C₄	C₂³	1	6	yes	—
61	Dic₁₃₉⋊C₄	C₂³	1	6	yes	—
62	Dic₁₃₉⋊C₄	C₂³	1	6	yes	—
63	Dic₁₃₉⋊C₄	C₂³	1	12	yes	—
64	C₂²⋊Dic₁₃₉	C₂³	1	12	yes	—
65	C₄×Dic₁₃₉	C₂³	1	2	yes	—
66	C₄×Dic₁₃₉	C₂³	1	6	yes	—
67	C₄×Dic₁₃₉	C₂³	1	6	yes	—
68	C₄×Dic₁₃₉	C₂³	1	6	yes	—
69	#41×C₁₃₉	Q₈	1	8	yes	complemented
70	C₂×Q₈×C₁₃₉	C₂³	1	1	yes	characteristic
71	C₂×D₄×C₁₃₉	C₂³	1	3	yes	—
72	C₂²×C₄×C₁₃₉	C₂³	1	3	yes	—
73	C₂²×C₄×C₁₃₉	C₂³	1	3	yes	—
74	C₂²×C₄×C₁₃₉	C₂³	1	3	yes	—
75	C₄⋊C₄×C₁₃₉	C₂³	1	3	yes	—
76	C₄⋊C₄×C₁₃₉	C₂³	1	3	yes	—
77	C₂²⋊C₄×C₁₃₉	C₂³	1	6	yes	—
78	C₄²×C₁₃₉	C₂³	1	1	yes	characteristic
79	C₄²×C₁₃₉	C₂³	1	3	yes	—
80	C₄²×C₁₃₉	C₂³	1	3	yes	—
81	C₄²×C₁₃₉	C₂³	1	3	yes	—
82	Dic₂₇₈	—	2	6	no	—
83	Dic₂₇₈	—	2	12	no	—
84	Dic₂₇₈	—	4	6	no	—
85	Dic₂₇₈	#41	1	4	yes	direct product factor
86	C₂×Dic₁₃₉	—	2	12	no	—
87	C₂×Dic₁₃₉	—	4	6	no	—
88	C₂×Dic₁₃₉	C₂⁴	1	2	yes	—
89	C₂×Dic₁₃₉	C₂⁴	1	6	yes	—
90	C₂×Dic₁₃₉	C₂⁴	1	6	yes	—
91	C₂×Dic₁₃₉	C₂⁴	1	2	yes	—
92	C₂³×C₁₃₉	C₂⁴	1	3	yes	—
93	Q₈×C₁₃₉	C₂×Q₈	1	4	yes	complemented
94	D₄×C₁₃₉	C₂×Q₈	1	12	yes	complemented
95	C₂×C₄×C₁₃₉	—	2	6	no	—
96	C₂×C₄×C₁₃₉	—	4	3	no	—
97	C₂×C₄×C₁₃₉	C₂×Q₈	1	12	yes	complemented
98	C₂×C₄×C₁₃₉	C₂⁴	1	1	yes	characteristic
99	C₂×C₄×C₁₃₉	C₂⁴	1	3	yes	—
100	C₂×C₄×C₁₃₉	C₂⁴	1	3	yes	—
101	C₂×C₄×C₁₃₉	C₂⁴	1	3	yes	—
102	C₂×C₄×C₁₃₉	C₂⁴	1	1	yes	characteristic
103	C₂×C₄×C₁₃₉	C₂⁴	1	1	yes	characteristic
104	C₂²×C₁₃₉	—	2	3	no	—
105	C₂²×C₁₃₉	—	2	3	no	—
106	C₂²×C₁₃₉	C₂²×Q₈	1	6	yes	—
107	C₂²×C₁₃₉	C₂⁵	1	1	yes	characteristic
108	C₄×C₁₃₉	—	2	3	no	—
109	C₄×C₁₃₉	—	2	1	no	—
110	C₄×C₁₃₉	—	2	3	no	—
111	C₄×C₁₃₉	C₂²×Q₈	1	2	yes	—
112	C₄×C₁₃₉	C₂²×Q₈	1	6	yes	—
113	C₄×C₁₃₉	C₂×#41	1	2	yes	—
114	Dic₁₃₉	—	2	6	no	—
115	Dic₁₃₉	—	2	2	no	—
116	Dic₁₃₉	—	2	6	no	—
117	Dic₁₃₉	C₂×#41	1	4	yes	—
118	C₂×C₁₃₉	—	2	6	no	—
119	C₂×C₁₃₉	C₂³×Q₈	1	1	yes	characteristic
120	C₂×C₁₃₉	C₂²×#41	1	1	yes	characteristic
121	C₂×C₁₃₉	#585	1	1	yes	characteristic
122	C₁₃₉	Q₈×#41	1	1	yes	characteristic, complemented, Sylow
123	Q₈×#41	—	139	1	no	maximal, Sylow
124	Q₈²	—	139	1	no	—
125	#557	—	139	2	no	—
126	#557	—	139	1	no	—
127	#554	—	139	3	no	—
128	#554	—	139	6	no	—
129	D₄×Q₈	—	139	3	no	—
130	#533	—	139	6	no	—
131	#533	—	139	3	no	—
132	C₄×#41	—	139	2	no	—
133	C₄×#41	D₁₃₉	1	1	yes	characteristic
134	C₂×C₄×Q₈	—	139	3	no	—
135	C₂×#41	C₂×D₁₃₉	1	1	yes	characteristic
136	C₂²×Q₈	—	139	3	no	—
137	C₄⋊Q₈	—	139	6	no	—
138	C₄⋊Q₈	—	139	6	no	—
139	C₄⋊Q₈	—	139	3	no	—
140	C₄⋊Q₈	—	139	3	no	—
141	#125	—	139	6	no	—
142	#125	—	139	6	no	—
143	#125	—	139	6	no	—
144	C₂²⋊Q₈	—	139	6	no	—
145	C₂²⋊Q₈	—	139	6	no	—
146	C₂²⋊Q₈	—	139	12	no	—
147	C₂²⋊Q₈	—	139	12	no	—
148	C₄×Q₈	—	139	2	no	—
149	C₄×Q₈	—	139	2	no	—
150	C₄×Q₈	—	139	1	no	—
151	C₄×Q₈	—	139	6	no	—
152	C₄×Q₈	—	139	3	no	—
153	C₄×Q₈	—	139	3	no	—
154	C₄×Q₈	—	139	3	no	—
155	C₄×Q₈	—	139	6	no	—
156	C₄×Q₈	—	139	6	no	—
157	C₄×Q₈	C₂×D₁₃₉	1	1	yes	characteristic
158	C₄×Q₈	C₂×D₁₃₉	1	1	yes	characteristic
159	C₄×D₄	—	139	6	no	—
160	C₄×D₄	—	139	6	no	—
161	C₄×D₄	C₂×D₁₃₉	1	3	yes	—
162	C₄×D₄	C₂×D₁₃₉	1	3	yes	—
163	#117	—	139	6	no	—
164	#117	C₂×D₁₃₉	1	3	yes	—
165	C₂×C₄⋊C₄	—	139	6	no	—
166	C₂×C₄⋊C₄	—	139	3	no	—
167	C₂×C₄²	—	139	6	no	—
168	C₂×C₄²	C₂×D₁₃₉	1	3	yes	—
169	#41	Dic₂₇₈	1	8	yes	direct product factor
170	C₂×Q₈	—	278	12	no	—
171	C₂×Q₈	—	139	3	no	—
172	C₂×Q₈	—	139	6	no	—
173	C₂×Q₈	—	139	6	no	—
174	C₂×Q₈	—	139	1	no	—
175	C₂×Q₈	C₂²×D₁₃₉	1	1	yes	characteristic
176	C₂×D₄	C₂²×D₁₃₉	1	3	yes	—
177	C₂²×C₄	—	139	6	no	—
178	C₂²×C₄	—	139	6	no	—
179	C₂²×C₄	C₂²×D₁₃₉	1	3	yes	—
180	C₂²×C₄	C₂²×D₁₃₉	1	3	yes	—
181	C₂²×C₄	C₂²×D₁₃₉	1	3	yes	—
182	C₄⋊C₄	—	139	2	no	—
183	C₄⋊C₄	—	139	1	no	—
184	C₄⋊C₄	—	139	6	no	—
185	C₄⋊C₄	—	139	6	no	—
186	C₄⋊C₄	—	139	6	no	—
187	C₄⋊C₄	—	139	6	no	—
188	C₄⋊C₄	—	139	3	no	—
189	C₄⋊C₄	—	139	12	no	—
190	C₄⋊C₄	—	139	6	no	—
191	C₄⋊C₄	—	139	6	no	—
192	C₄⋊C₄	—	139	6	no	—
193	C₄⋊C₄	C₂²×D₁₃₉	1	3	yes	—
194	C₄⋊C₄	C₂²×D₁₃₉	1	3	yes	—
195	C₂²⋊C₄	—	139	12	no	—
196	C₂²⋊C₄	C₂²×D₁₃₉	1	6	yes	—
197	C₄²	—	139	6	no	—
198	C₄²	—	139	6	no	—
199	C₄²	—	139	2	no	—
200	C₄²	—	139	6	no	—
201	C₄²	C₂²×D₁₃₉	1	3	yes	—
202	C₄²	C₂²×D₁₃₉	1	3	yes	—
203	C₄²	C₂²×D₁₃₉	1	1	yes	characteristic
204	C₄²	C₂²×D₁₃₉	1	3	yes	—
205	C₂³	C₂³×D₁₃₉	1	3	yes	—
206	Q₈	—	278	12	no	—
207	Q₈	—	278	6	no	—
208	Q₈	—	556	6	no	—
209	Q₈	—	139	4	no	—
210	Q₈	C₂×Dic₂₇₈	1	4	yes	complemented
211	D₄	C₂×Dic₂₇₈	1	12	yes	complemented
212	C₂×C₄	—	2	6	no	—
213	C₂×C₄	—	4	3	no	—
214	C₂×C₄	—	278	12	no	—
215	C₂×C₄	—	556	6	no	—
216	C₂×C₄	—	139	2	no	—
217	C₂×C₄	—	139	6	no	—
218	C₂×C₄	—	139	6	no	—
219	C₂×C₄	—	139	2	no	—
220	C₂×C₄	C₂×Dic₂₇₈	1	12	yes	complemented
221	C₂×C₄	C₂³×D₁₃₉	1	1	yes	characteristic
222	C₂×C₄	C₂³×D₁₃₉	1	3	yes	—
223	C₂×C₄	C₂³×D₁₃₉	1	3	yes	—
224	C₂×C₄	C₂³×D₁₃₉	1	3	yes	—
225	C₂×C₄	C₂³×D₁₃₉	1	1	yes	characteristic
226	C₂×C₄	C₂³×D₁₃₉	1	1	yes	characteristic
227	C₂²	—	2	3	no	—
228	C₂²	—	2	3	no	—
229	C₂²	C₂²×Dic₂₇₈	1	6	yes	—
230	C₂²	C₂⁴×D₁₃₉	1	1	yes	characteristic
231	C₄	—	2	3	no	—
232	C₄	—	2	1	no	—
233	C₄	—	2	3	no	—
234	C₄	—	278	6	no	—
235	C₄	—	278	2	no	—
236	C₄	—	278	6	no	—
237	C₄	—	139	4	no	—
238	C₄	C₂²×Dic₂₇₈	1	2	yes	—
239	C₄	C₂²×Dic₂₇₈	1	6	yes	—
240	C₄	#41×D₁₃₉	1	2	yes	—
241	C₂	—	2	6	no	—
242	C₂	C₂³×Dic₂₇₈	1	1	yes	characteristic
243	C₂	C₂×#41×D₁₃₉	1	1	yes	characteristic
244	C₂	#22581789	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂³×D₁₃₉
commutator, socle, cosocle	C₂²×C₁₃₉	C₂⁵
Frattini	C₂²	C₂⁴×D₁₃₉
Fitting	C₄×#41×C₁₃₉	C₂
prime radical	C₂×#41×C₁₃₉	C₂²
hypercenter	C₄×#41	D₁₃₉
hypocenter	C₁₃₉	Q₈×#41
LCS	G ▷ C₂²×C₁₃₉ ▷ C₁₃₉
UCS	C₁ ◁ C₂×C₄ ◁ C₄×#41

Complements

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

N	H ≅ G/N	type
1. Q₈×Dic₂₇₈	C₂	semidirect
2. #22581682	C₂	semidirect
3. #22581679	C₂	semidirect
4. #22581662	C₂	semidirect
5. #22581660	C₂	semidirect
6. D₄×Dic₂₇₈	C₂	semidirect
7. #22581510	C₂	semidirect
8. #22581506	C₂	semidirect
10. C₂×C₄×Dic₂₇₈	C₂	semidirect
69. #41×C₁₃₉	Q₈	semidirect
85. Dic₂₇₈	#41	direct (N × H)
93. Q₈×C₁₃₉	C₂×Q₈	semidirect
94. D₄×C₁₃₉	C₂×Q₈	semidirect
97. C₂×C₄×C₁₃₉	C₂×Q₈	semidirect
122. C₁₃₉	Q₈×#41	semidirect
169. #41	Dic₂₇₈	direct (N × H)
210. Q₈	C₂×Dic₂₇₈	semidirect
211. D₄	C₂×Dic₂₇₈	semidirect
220. C₂×C₄	C₂×Dic₂₇₈	semidirect

Maximal subgroups

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

subgroup	index
Q₈×Dic₂₇₈	2
#22581682	2
#22581679	2
#22581662	2
#22581660	2
D₄×Dic₂₇₈	2
#22581510	2
#22581506	2
#41×Dic₁₃₉	2
C₂×C₄×Dic₂₇₈	2
C₄×#41×C₁₃₉	2
Q₈×#41	139

Subgroups — #41×Dic278 #31819984

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #41×Dic₂₇₈ #31819984