Group #31819118 — subgroups

Subgroups — #31819118

Order 17792.

372 conjugacy classes of subgroups; only the 123 normal subgroups are shown.

normal subgroups only

This group has 372 conjugacy classes of subgroups — too many to draw. Only its 123 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	Q₈×D₅₅₆	C₂	1	1	yes	maximal, characteristic, complemented
2	#22581312	C₂	1	2	yes	maximal, complemented
3	#22581303	C₂	1	1	yes	maximal, characteristic, complemented
4	#22581248	C₂	1	2	yes	maximal, complemented
5	#22581233	C₂	1	1	yes	maximal, characteristic, complemented
6	#22581165	C₂	1	2	yes	maximal
7	#22581155	C₂	1	1	yes	maximal, characteristic, complemented
8	#22581084	C₂	1	1	yes	maximal, characteristic, complemented
9	#22581051	C₂	1	2	yes	maximal, complemented
10	C₄×#14555770	C₂	1	1	yes	maximal, characteristic
11	#556×C₁₃₉	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×Q₈×D₁₃₉	—	2	1	no	—
13	C₂×#14555772	C₂²	1	2	yes	—
14	C₂×#14555770	C₂²	1	1	yes	characteristic
15	#17984842	C₂²	1	2	yes	—
16	#17984842	C₂²	1	1	yes	characteristic
17	#17984822	—	2	1	no	—
18	#17984822	—	2	2	no	—
19	C₄×D₅₅₆	C₂²	1	1	yes	characteristic
20	C₄×D₅₅₆	C₂²	1	2	yes	—
21	#17984775	C₂²	1	1	yes	characteristic
22	#17984775	C₂²	1	2	yes	—
23	C₄⋊(C₁₃₉⋊C₈)	C₂²	1	2	yes	—
24	C₄⋊(C₁₃₉⋊C₈)	C₂²	1	1	yes	characteristic
25	#17984766	C₂²	1	2	yes	—
26	#17984766	C₂²	1	2	yes	—
27	#17984766	C₂²	1	2	yes	—
28	#17984763	C₂²	1	1	yes	characteristic
29	#17984763	C₂²	1	2	yes	—
30	#17984762	C₂²	1	1	yes	characteristic
31	#17984753	C₂²	1	2	yes	—
32	C₄×C₁₃₉⋊C₈	C₂²	1	1	yes	characteristic
33	C₄⋊Q₈×C₁₃₉	C₂²	1	1	yes	characteristic
34	#126×C₁₃₉	—	2	2	no	—
35	#124×C₁₃₉	C₂²	1	2	yes	—
36	C₂²⋊Q₈×C₁₃₉	—	2	1	no	—
37	C₄×Q₈×C₁₃₉	C₂²	1	2	yes	—
38	C₄×Q₈×C₁₃₉	C₂²	1	1	yes	characteristic
39	C₄×D₄×C₁₃₉	C₂²	1	1	yes	characteristic
40	#117×C₁₃₉	—	2	1	no	—
41	C₂×Dic₂₇₈	—	2	2	no	—
42	C₂×Dic₂₇₈	—	2	1	no	—
43	Q₈×D₁₃₉	—	4	4	no	—
44	C₂×D₅₅₆	C₂³	1	1	yes	characteristic
45	C₂×C₄×D₁₃₉	—	2	2	no	—
46	C₂×C₄×D₁₃₉	—	2	1	no	—
47	#14555772	—	2	4	no	—
48	#14555770	—	2	2	no	—
49	C₂×C₁₃₉⋊C₈	C₂³	1	1	yes	characteristic
50	C₂×C₁₃₉⋊C₈	C₂³	1	2	yes	—
51	C₂×C₁₃₉⋊C₈	C₂³	1	1	yes	characteristic
52	C₄⋊Dic₁₃₉	—	2	2	no	—
53	C₄⋊Dic₁₃₉	—	2	1	no	—
54	C₄⋊Dic₁₃₉	C₂³	1	1	yes	characteristic
55	C₄⋊Dic₁₃₉	C₂³	1	2	yes	—
56	(C₂×D₁₃₉)⋊C₄	—	2	2	no	—
57	(C₂×D₁₃₉)⋊C₄	—	2	1	no	—
58	C₂×Q₈×C₁₃₉	D₄	1	1	yes	characteristic
59	C₂×Q₈×C₁₃₉	C₂³	1	2	yes	—
60	C₂×D₄×C₁₃₉	C₂³	1	1	yes	characteristic
61	C₂²×C₄×C₁₃₉	—	2	1	no	—
62	C₄⋊C₄×C₁₃₉	—	2	1	no	—
63	C₄⋊C₄×C₁₃₉	—	2	2	no	—
64	C₄⋊C₄×C₁₃₉	—	2	1	no	—
65	C₄⋊C₄×C₁₃₉	D₄	1	1	yes	characteristic
66	C₄⋊C₄×C₁₃₉	D₄	1	2	yes	—
67	C₄⋊C₄×C₁₃₉	C₂³	1	1	yes	characteristic
68	C₂²⋊C₄×C₁₃₉	—	2	2	no	—
69	C₂²⋊C₄×C₁₃₉	—	2	1	no	—
70	C₂²⋊C₄×C₁₃₉	—	2	2	no	—
71	C₄²×C₁₃₉	—	2	2	no	—
72	C₄²×C₁₃₉	C₂³	1	1	yes	characteristic
73	C₄²×C₁₃₉	C₂³	1	2	yes	—
74	C₂²×D₁₃₉	—	2	1	no	—
75	Dic₂₇₈	—	4	4	no	—
76	Dic₂₇₈	—	4	2	no	—
77	D₅₅₆	—	2	1	no	—
78	D₅₅₆	#41	1	2	yes	—
79	C₂×Dic₁₃₉	—	2	2	no	—
80	C₂×Dic₁₃₉	—	2	1	no	—
81	C₄×D₁₃₉	—	4	4	no	—
82	C₄×D₁₃₉	—	4	2	no	—
83	C₁₃₉⋊C₈	—	2	1	no	—
84	C₁₃₉⋊C₈	—	2	2	no	—
85	C₁₃₉⋊C₈	—	2	1	no	—
86	C₂³×C₁₃₉	—	2	1	no	—
87	Q₈×C₁₃₉	—	2	2	no	—
88	Q₈×C₁₃₉	—	2	2	no	—
89	Q₈×C₁₃₉	—	2	2	no	—
90	D₄×C₁₃₉	—	2	1	no	—
91	D₄×C₁₃₉	—	2	1	no	—
92	C₂×C₄×C₁₃₉	—	2	1	no	—
93	C₂×C₄×C₁₃₉	—	2	2	no	—
94	C₂×C₄×C₁₃₉	—	4	2	no	—
95	C₂×C₄×C₁₃₉	C₂×D₄	1	2	yes	—
96	C₂×C₄×C₁₃₉	C₂×D₄	1	1	yes	characteristic
97	C₂×C₄×C₁₃₉	C₂×D₄	1	2	yes	—
98	C₂×C₄×C₁₃₉	C₂×D₄	1	1	yes	characteristic
99	C₂×C₄×C₁₃₉	C₂⁴	1	1	yes	characteristic
100	C₂²×C₁₃₉	—	4	1	no	—
101	C₂²×C₁₃₉	—	4	1	no	—
102	C₂²×C₁₃₉	—	4	1	no	—
103	C₂²×C₁₃₉	C₂²×D₄	1	1	yes	characteristic
104	C₂×D₁₃₉	—	2	2	no	—
105	C₂×D₁₃₉	—	4	1	no	—
106	C₂×D₁₃₉	—	4	1	no	—
107	C₄×C₁₃₉	—	2	2	no	—
108	C₄×C₁₃₉	—	2	2	no	—
109	C₄×C₁₃₉	—	2	1	no	—
110	C₄×C₁₃₉	—	4	2	no	—
111	C₄×C₁₃₉	—	4	1	no	—
112	C₄×C₁₃₉	AGL(1,ℤ/8)	1	2	yes	—
113	C₄×C₁₃₉	C₂×#41	1	1	yes	characteristic
114	C₄×C₁₃₉	2₋¹⁺⁴	1	1	yes	characteristic
115	Dic₁₃₉	—	4	2	no	—
116	Dic₁₃₉	—	4	1	no	—
117	C₂×C₁₃₉	—	8	1	no	—
118	C₂×C₁₃₉	#548	1	1	yes	characteristic
119	C₂×C₁₃₉	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
120	C₂×C₁₃₉	#578	1	1	yes	characteristic
121	D₁₃₉	—	4	2	no	—
122	C₁₃₉	#3367	1	1	yes	characteristic, complemented, Sylow
123	#3367	—	139	1	no	maximal, Sylow
124	#556	D₁₃₉	1	1	yes	characteristic, complemented
125	D₄×Q₈	—	139	1	no	—
126	#489	—	139	2	no	—
127	#487	—	139	1	no	—
128	#477	—	139	2	no	—
129	#474	—	139	1	no	—
130	#463	—	139	2	no	—
131	#461	—	139	1	no	—
132	#446	—	139	1	no	—
133	#440	—	139	2	no	—
134	C₄×D₈	—	139	1	no	—
135	C₂²×Q₈	—	278	1	no	—
136	C₂×SD₁₆	—	139	2	no	—
137	C₂×D₈	—	139	1	no	—
138	C₄⋊Q₈	—	139	1	no	—
139	C₄⋊Q₈	—	139	2	no	—
140	C₄⋊Q₈	C₂×D₁₃₉	1	1	yes	characteristic
141	#126	—	2	2	no	—
142	#124	C₂×D₁₃₉	1	2	yes	—
143	C₂²⋊Q₈	—	2	1	no	—
144	C₂²⋊Q₈	—	278	1	no	—
145	C₂²⋊Q₈	—	278	2	no	—
146	C₄×Q₈	C₂×D₁₃₉	1	2	yes	—
147	C₄×Q₈	C₂×D₁₃₉	1	1	yes	characteristic
148	C₄×D₄	—	139	1	no	—
149	C₄×D₄	—	139	2	no	—
150	C₄×D₄	C₂×D₁₃₉	1	1	yes	characteristic
151	#117	—	2	1	no	—
152	#107	—	139	1	no	—
153	#107	—	139	2	no	—
154	C₄⋊C₈	—	139	1	no	—
155	C₄⋊C₈	—	139	2	no	—
156	#103	—	139	2	no	—
157	#103	—	139	2	no	—
158	#103	—	139	2	no	—
159	#102	—	139	1	no	—
160	#102	—	139	2	no	—
161	#102	—	139	1	no	—
162	#97	—	139	2	no	—
163	C₄×C₈	—	139	1	no	—
164	C₂×Q₈	—	278	2	no	—
165	C₂×Q₈	—	278	1	no	—
166	C₂×Q₈	—	556	4	no	—
167	C₂×Q₈	C₂²⋊D₁₃₉	1	1	yes	characteristic
168	C₂×Q₈	C₂²×D₁₃₉	1	2	yes	—
169	C₂×D₄	—	139	1	no	—
170	C₂×D₄	C₂²×D₁₃₉	1	1	yes	characteristic
171	C₂²×C₄	—	2	1	no	—
172	C₂²×C₄	—	278	2	no	—
173	C₂²×C₄	—	278	1	no	—
174	SD₁₆	—	278	4	no	—
175	D₈	—	278	2	no	—
176	C₂×C₈	—	139	1	no	—
177	C₂×C₈	—	139	1	no	—
178	C₂×C₈	—	139	2	no	—
179	C₄⋊C₄	—	2	1	no	—
180	C₄⋊C₄	—	2	2	no	—
181	C₄⋊C₄	—	2	1	no	—
182	C₄⋊C₄	—	278	2	no	—
183	C₄⋊C₄	—	278	1	no	—
184	C₄⋊C₄	—	139	1	no	—
185	C₄⋊C₄	—	139	2	no	—
186	C₄⋊C₄	C₂²⋊D₁₃₉	1	1	yes	characteristic
187	C₄⋊C₄	C₂²⋊D₁₃₉	1	2	yes	—
188	C₄⋊C₄	C₂²×D₁₃₉	1	1	yes	characteristic
189	C₂²⋊C₄	—	2	2	no	—
190	C₂²⋊C₄	—	2	1	no	—
191	C₂²⋊C₄	—	2	2	no	—
192	C₂²⋊C₄	—	278	2	no	—
193	C₂²⋊C₄	—	278	1	no	—
194	C₄²	—	2	2	no	—
195	C₄²	C₂²×D₁₃₉	1	1	yes	characteristic
196	C₄²	C₂²×D₁₃₉	1	2	yes	—
197	C₂³	—	2	1	no	—
198	C₂³	—	278	1	no	—
199	Q₈	—	2	2	no	—
200	Q₈	—	2	2	no	—
201	Q₈	—	2	2	no	—
202	Q₈	—	556	4	no	—
203	Q₈	—	556	2	no	—
204	D₄	—	2	1	no	—
205	D₄	—	2	1	no	—
206	D₄	—	278	1	no	—
207	D₄	—	139	2	no	—
208	C₂×C₄	—	2	1	no	—
209	C₂×C₄	—	2	2	no	—
210	C₂×C₄	—	4	2	no	—
211	C₂×C₄	—	278	2	no	—
212	C₂×C₄	—	278	1	no	—
213	C₂×C₄	—	556	4	no	—
214	C₂×C₄	—	556	2	no	—
215	C₂×C₄	C₂×C₂²⋊D₁₃₉	1	2	yes	—
216	C₂×C₄	C₂×C₂²⋊D₁₃₉	1	1	yes	characteristic
217	C₂×C₄	C₂×C₂²⋊D₁₃₉	1	2	yes	—
218	C₂×C₄	C₂×C₂²⋊D₁₃₉	1	1	yes	characteristic
219	C₂×C₄	C₂³×D₁₃₉	1	1	yes	characteristic
220	C₈	—	278	1	no	—
221	C₈	—	278	2	no	—
222	C₈	—	278	1	no	—
223	C₂²	—	4	1	no	—
224	C₂²	—	4	1	no	—
225	C₂²	—	4	1	no	—
226	C₂²	—	278	2	no	—
227	C₂²	—	556	1	no	—
228	C₂²	—	556	1	no	—
229	C₂²	C₂²×C₂²⋊D₁₃₉	1	1	yes	characteristic
230	C₄	—	2	2	no	—
231	C₄	—	2	1	no	—
232	C₄	—	2	2	no	—
233	C₄	—	4	2	no	—
234	C₄	—	4	1	no	—
235	C₄	—	556	2	no	—
236	C₄	—	556	1	no	—
237	C₄	#17984867	1	2	yes	—
238	C₄	#41×D₁₃₉	1	1	yes	characteristic
239	C₄	#17984889	1	1	yes	characteristic
240	C₂	—	8	1	no	—
241	C₂	—	556	2	no	—
242	C₂	#22581610	1	1	yes	characteristic
243	C₂	C₂×#17984867	1	1	yes	characteristic
244	C₂	#22581768	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

Q₈×D₅₅₆

C₂

yes

maximal, characteristic, complemented

#22581312

C₂

yes

maximal, complemented

#22581303

C₂

yes

maximal, characteristic, complemented

#22581248

C₂

yes

maximal, complemented

#22581233

C₂

yes

maximal, characteristic, complemented

#22581165

C₂

yes

maximal

#22581155

C₂

yes

maximal, characteristic, complemented

#22581084

C₂

yes

maximal, characteristic, complemented

#22581051

C₂

yes

maximal, complemented

C₄×#14555770

C₂

yes

maximal, characteristic

#556×C₁₃₉

C₂

yes

maximal, characteristic, complemented

C₂×Q₈×D₁₃₉

—

C₂×#14555772

C₂²

yes

—

C₂×#14555770

C₂²

yes

characteristic

#17984842

C₂²

yes

—

#17984842

C₂²

yes

characteristic

#17984822

—

#17984822

—

C₄×D₅₅₆

C₂²

yes

characteristic

C₄×D₅₅₆

C₂²

yes

—

#17984775

C₂²

yes

characteristic

#17984775

C₂²

yes

—

C₄⋊(C₁₃₉⋊C₈)

C₂²

yes

—

C₄⋊(C₁₃₉⋊C₈)

C₂²

yes

characteristic

#17984766

C₂²

yes

—

#17984766

C₂²

yes

—

#17984766

C₂²

yes

—

#17984763

C₂²

yes

characteristic

#17984763

C₂²

yes

—

#17984762

C₂²

yes

characteristic

#17984753

C₂²

yes

—

C₄×C₁₃₉⋊C₈

C₂²

yes

characteristic

C₄⋊Q₈×C₁₃₉

C₂²

yes

characteristic

#126×C₁₃₉

—

#124×C₁₃₉

C₂²

yes

—

C₂²⋊Q₈×C₁₃₉

—

C₄×Q₈×C₁₃₉

C₂²

yes

—

C₄×Q₈×C₁₃₉

C₂²

yes

characteristic

C₄×D₄×C₁₃₉

C₂²

yes

characteristic

#117×C₁₃₉

—

C₂×Dic₂₇₈

—

C₂×Dic₂₇₈

—

Q₈×D₁₃₉

—

C₂×D₅₅₆

C₂³

yes

characteristic

C₂×C₄×D₁₃₉

—

C₂×C₄×D₁₃₉

—

#14555772

—

#14555770

—

C₂×C₁₃₉⋊C₈

C₂³

yes

characteristic

C₂×C₁₃₉⋊C₈

C₂³

yes

—

C₂×C₁₃₉⋊C₈

C₂³

yes

characteristic

C₄⋊Dic₁₃₉

—

C₄⋊Dic₁₃₉

—

C₄⋊Dic₁₃₉

C₂³

yes

characteristic

C₄⋊Dic₁₃₉

C₂³

yes

—

(C₂×D₁₃₉)⋊C₄

—

(C₂×D₁₃₉)⋊C₄

—

C₂×Q₈×C₁₃₉

D₄

yes

characteristic

C₂×Q₈×C₁₃₉

C₂³

yes

—

C₂×D₄×C₁₃₉

C₂³

yes

characteristic

C₂²×C₄×C₁₃₉

—

C₄⋊C₄×C₁₃₉

—

C₄⋊C₄×C₁₃₉

—

C₄⋊C₄×C₁₃₉

—

C₄⋊C₄×C₁₃₉

D₄

yes

characteristic

C₄⋊C₄×C₁₃₉

D₄

yes

—

C₄⋊C₄×C₁₃₉

C₂³

yes

characteristic

C₂²⋊C₄×C₁₃₉

—

C₂²⋊C₄×C₁₃₉

—

C₂²⋊C₄×C₁₃₉

—

C₄²×C₁₃₉

—

C₄²×C₁₃₉

C₂³

yes

characteristic

C₄²×C₁₃₉

C₂³

yes

—

C₂²×D₁₃₉

—

Dic₂₇₈

—

Dic₂₇₈

—

D₅₅₆

—

D₅₅₆

#41

yes

—

C₂×Dic₁₃₉

—

C₂×Dic₁₃₉

—

C₄×D₁₃₉

—

C₄×D₁₃₉

—

C₁₃₉⋊C₈

—

C₁₃₉⋊C₈

—

C₁₃₉⋊C₈

—

C₂³×C₁₃₉

—

Q₈×C₁₃₉

—

Q₈×C₁₃₉

—

Q₈×C₁₃₉

—

D₄×C₁₃₉

—

D₄×C₁₃₉

—

C₂×C₄×C₁₃₉

—

C₂×C₄×C₁₃₉

—

C₂×C₄×C₁₃₉

—

C₂×C₄×C₁₃₉

C₂×D₄

yes

—

C₂×C₄×C₁₃₉

C₂×D₄

yes

characteristic

C₂×C₄×C₁₃₉

C₂×D₄

yes

—

C₂×C₄×C₁₃₉

C₂×D₄

yes

characteristic

C₂×C₄×C₁₃₉

C₂⁴

yes

characteristic

100

C₂²×C₁₃₉

—

101

C₂²×C₁₃₉

—

102

C₂²×C₁₃₉

—

103

C₂²×C₁₃₉

C₂²×D₄

yes

characteristic

104

C₂×D₁₃₉

—

105

C₂×D₁₃₉

—

106

C₂×D₁₃₉

—

107

C₄×C₁₃₉

—

108

C₄×C₁₃₉

—

109

C₄×C₁₃₉

—

110

C₄×C₁₃₉

—

111

C₄×C₁₃₉

—

112

C₄×C₁₃₉

AGL(1,ℤ/8)

yes

—

113

C₄×C₁₃₉

C₂×#41

yes

characteristic

114

C₄×C₁₃₉

2₋¹⁺⁴

yes

characteristic

115

Dic₁₃₉

—

116

Dic₁₃₉

—

117

C₂×C₁₃₉

—

118

C₂×C₁₃₉

#548

yes

characteristic

119

C₂×C₁₃₉

C₂×AGL(1,ℤ/8)

yes

characteristic

120

C₂×C₁₃₉

#578

yes

characteristic

121

D₁₃₉

—

122

C₁₃₉

#3367

yes

characteristic, complemented, Sylow

123

#3367

—

139

maximal, Sylow

124

#556

D₁₃₉

yes

characteristic, complemented

125

D₄×Q₈

—

139

—

126

#489

—

139

—

127

#487

—

139

—

128

#477

—

139

—

129

#474

—

139

—

130

#463

—

139

—

131

#461

—

139

—

132

#446

—

139

—

133

#440

—

139

—

134

C₄×D₈

—

139

—

135

C₂²×Q₈

—

278

—

136

C₂×SD₁₆

—

139

—

137

C₂×D₈

—

139

—

138

C₄⋊Q₈

—

139

—

139

C₄⋊Q₈

—

139

—

140

C₄⋊Q₈

C₂×D₁₃₉

yes

characteristic

141

#126

—

142

#124

C₂×D₁₃₉

yes

—

143

C₂²⋊Q₈

—

144

C₂²⋊Q₈

—

278

—

145

C₂²⋊Q₈

—

278

—

146

C₄×Q₈

C₂×D₁₃₉

yes

—

147

C₄×Q₈

C₂×D₁₃₉

yes

characteristic

148

C₄×D₄

—

139

—

149

C₄×D₄

—

139

—

150

C₄×D₄

C₂×D₁₃₉

yes

characteristic

151

#117

—

152

#107

—

139

—

153

#107

—

139

—

154

C₄⋊C₈

—

139

—

155

C₄⋊C₈

—

139

—

156

#103

—

139

—

157

#103

—

139

—

158

#103

—

139

—

159

#102

—

139

—

160

#102

—

139

—

161

#102

—

139

—

162

#97

—

139

—

163

C₄×C₈

—

139

—

164

C₂×Q₈

—

278

—

165

C₂×Q₈

—

278

—

166

C₂×Q₈

—

556

—

167

C₂×Q₈

C₂²⋊D₁₃₉

yes

characteristic

168

C₂×Q₈

C₂²×D₁₃₉

yes

—

169

C₂×D₄

—

139

—

170

C₂×D₄

C₂²×D₁₃₉

yes

characteristic

171

C₂²×C₄

—

172

C₂²×C₄

—

278

—

173

C₂²×C₄

—

278

—

174

SD₁₆

—

278

—

175

D₈

—

278

—

176

C₂×C₈

—

139

—

177

C₂×C₈

—

139

—

178

C₂×C₈

—

139

—

179

C₄⋊C₄

—

180

C₄⋊C₄

—

181

C₄⋊C₄

—

182

C₄⋊C₄

—

278

—

183

C₄⋊C₄

—

278

—

184

C₄⋊C₄

—

139

—

185

C₄⋊C₄

—

139

—

186

C₄⋊C₄

C₂²⋊D₁₃₉

yes

characteristic

187

C₄⋊C₄

C₂²⋊D₁₃₉

yes

—

188

C₄⋊C₄

C₂²×D₁₃₉

yes

characteristic

189

C₂²⋊C₄

—

190

C₂²⋊C₄

—

191

C₂²⋊C₄

—

192

C₂²⋊C₄

—

278

—

193

C₂²⋊C₄

—

278

—

194

C₄²

—

195

C₄²

C₂²×D₁₃₉

yes

characteristic

196

C₄²

C₂²×D₁₃₉

yes

—

197

C₂³

—

198

C₂³

—

278

—

199

Q₈

—

200

Q₈

—

201

Q₈

—

202

Q₈

—

556

—

203

Q₈

—

556

—

204

D₄

—

205

D₄

—

206

D₄

—

278

—

207

D₄

—

139

—

208

C₂×C₄

—

209

C₂×C₄

—

210

C₂×C₄

—

211

C₂×C₄

—

278

—

212

C₂×C₄

—

278

—

213

C₂×C₄

—

556

—

214

C₂×C₄

—

556

—

215

C₂×C₄

C₂×C₂²⋊D₁₃₉

yes

—

216

C₂×C₄

C₂×C₂²⋊D₁₃₉

yes

characteristic

217

C₂×C₄

C₂×C₂²⋊D₁₃₉

yes

—

218

C₂×C₄

C₂×C₂²⋊D₁₃₉

yes

characteristic

219

C₂×C₄

C₂³×D₁₃₉

yes

characteristic

220

C₈

—

278

—

221

C₈

—

278

—

222

C₈

—

278

—

223

C₂²

—

224

C₂²

—

225

C₂²

—

226

C₂²

—

278

—

227

C₂²

—

556

—

228

C₂²

—

556

—

229

C₂²

C₂²×C₂²⋊D₁₃₉

yes

characteristic

230

C₄

—

231

C₄

—

232

C₄

—

233

C₄

—

234

C₄

—

235

C₄

—

556

—

236

C₄

—

556

—

237

C₄

#17984867

yes

—

238

C₄

#41×D₁₃₉

yes

characteristic

239

C₄

#17984889

yes

characteristic

240

C₂

—

241

C₂

—

556

—

242

C₂

#22581610

yes

characteristic

243

C₂

C₂×#17984867

yes

characteristic

244

C₂

#22581768

yes

characteristic

	subgroup	G/H
center	C₂²	C₂²×C₂²⋊D₁₃₉
commutator, cosocle	C₂×C₄×C₁₃₉	C₂⁴
Frattini	C₂×C₄	C₂³×D₁₃₉
Fitting	#556×C₁₃₉	C₂
socle	C₂²×C₁₃₉	C₂²×D₄
prime radical	C₂×#14555770	C₂²
hypercenter	#556	D₁₃₉
hypocenter	C₁₃₉	#3367
LCS	G ▷ C₂×C₄×C₁₃₉ ▷ C₂×C₁₃₉ ▷ C₁₃₉
UCS	C₁ ◁ C₂² ◁ C₄×Q₈ ◁ #556

subgroup

G/H

center

C₂²

C₂²×C₂²⋊D₁₃₉

commutator, cosocle

C₂×C₄×C₁₃₉

C₂⁴

Frattini

C₂×C₄

C₂³×D₁₃₉

Fitting

#556×C₁₃₉

C₂

socle

C₂²×C₁₃₉

C₂²×D₄

C₂²

D₁₃₉

C₁₃₉

G ▷ C₂×C₄×C₁₃₉ ▷ C₂×C₁₃₉ ▷ C₁₃₉

UCS

C₁ ◁ C₂² ◁ C₄×Q₈ ◁ #556

N	H ≅ G/N	type
1. Q₈×D₅₅₆	C₂	semidirect
2. #22581312	C₂	semidirect
3. #22581303	C₂	semidirect
4. #22581248	C₂	semidirect
5. #22581233	C₂	semidirect
7. #22581155	C₂	semidirect
8. #22581084	C₂	semidirect
9. #22581051	C₂	semidirect
11. #556×C₁₃₉	C₂	semidirect
122. C₁₃₉	#3367	semidirect
124. #556	D₁₃₉	semidirect

H ≅ G/N

type

1. Q₈×D₅₅₆

C₂

semidirect

2. #22581312

C₂

semidirect

3. #22581303

C₂

semidirect

4. #22581248

C₂

semidirect

5. #22581233

C₂

semidirect

7. #22581155

C₂

semidirect

8. #22581084

C₂

semidirect

9. #22581051

C₂

semidirect

11. #556×C₁₃₉

C₂

semidirect

122. C₁₃₉

#3367

semidirect

124. #556

D₁₃₉

semidirect

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

subgroup	index
Q₈×D₅₅₆	2
#22581312	2
#22581303	2
#22581248	2
#22581233	2
#22581165	2
#22581155	2
#22581084	2
#22581051	2
C₄×#14555770	2
#556×C₁₃₉	2
#3367	139

subgroup

index

Q₈×D₅₅₆

#556×C₁₃₉

139

Subgroups — #31819118

Conjugacy classes of subgroups

Complements

Maximal subgroups