Group #34444709 — subgroups

Subgroups — #34444709

Order 20224.

488 conjugacy classes of subgroups; only the 127 normal subgroups are shown.

normal subgroups only

This group has 488 conjugacy classes of subgroups — too many to draw. Only its 127 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	#23774040	C₂	1	2	yes	maximal, complemented
2	#23765550	C₂	1	2	yes	maximal, complemented
3	#23765536	C₂	1	2	yes	maximal, complemented
4	#23765515	C₂	1	2	yes	maximal, complemented
5	#23765486	C₂	1	1	yes	maximal, characteristic
6	#23765464	C₂	1	2	yes	maximal, complemented
7	#23765458	C₂	1	2	yes	maximal, complemented
8	#23765158	C₂	1	1	yes	maximal, characteristic, complemented
9	C₇₉×#2216	C₂	1	1	yes	maximal, characteristic, complemented
10	#19099186	—	2	2	no	—
11	C₂×#14579104	C₂²	1	2	yes	—
12	C₂×D₁₂₆₄	C₂²	1	1	yes	characteristic
13	C₂×#14579101	C₂²	1	1	yes	characteristic
14	#19098967	C₂²	1	2	yes	—
15	#19098948	C₂²	1	2	yes	—
16	#19098924	—	2	2	no	—
17	#19098811	—	2	2	no	—
18	#19098799	C₂²	1	2	yes	—
19	#19098666	—	2	2	no	—
20	Q₁₆×Dic₇₉	C₂²	1	2	yes	—
21	#19098645	C₂²	1	2	yes	—
22	#19098518	C₂²	1	1	yes	characteristic
23	#19098389	C₂²	1	1	yes	characteristic
24	#19098388	C₂²	1	2	yes	—
25	Dic₇₉⋊C₁₆	C₂²	1	2	yes	—
26	Dic₇₉⋊C₁₆	C₂²	1	2	yes	—
27	#19098367	C₂²	1	2	yes	—
28	#19098365	C₂²	1	2	yes	—
29	#19098363	C₂²	1	2	yes	—
30	C₂×Q₃₂×C₇₉	C₂²	1	1	yes	characteristic
31	C₂²×C₁₆×C₇₉	C₂²	1	1	yes	characteristic
32	#467×C₇₉	C₂²	1	2	yes	—
33	#366×C₇₉	C₂²	1	1	yes	characteristic
34	#358×C₇₉	C₂²	1	2	yes	—
35	C₂×#11929611	—	2	2	no	—
36	C₂×D₆₃₂	C₂³	1	2	yes	—
37	C₂×#11929608	D₄	1	1	yes	characteristic
38	#14579160	—	2	2	no	—
39	#14579156	—	4	2	no	—
40	#14579150	—	2	2	no	—
41	#14579147	—	4	2	no	—
42	#14579143	—	4	2	no	—
43	#14579141	—	4	2	no	—
44	#14579135	—	2	2	no	—
45	Q₈×Dic₇₉	—	2	2	no	—
46	#14579123	—	2	2	no	—
47	#14579120	D₄	1	1	yes	characteristic
48	#14579104	—	4	2	no	—
49	D₁₂₆₄	—	4	1	no	—
50	#14579101	—	4	1	no	—
51	C₂×C₇₉⋊C₁₆	C₂³	1	2	yes	—
52	#14579095	C₂³	1	2	yes	—
53	#14579094	—	2	2	no	—
54	#14579085	—	2	2	no	—
55	#14579083	—	2	2	no	—
56	#14579081	—	2	2	no	—
57	#14579071	D₄	1	2	yes	—
58	C₈×Dic₇₉	C₂³	1	2	yes	—
59	C₂×Q₁₆×C₇₉	C₂³	1	2	yes	—
60	C₂²×C₈×C₇₉	C₂³	1	1	yes	characteristic
61	C₂²⋊Q₈×C₇₉	—	2	2	no	—
62	Q₃₂×C₇₉	—	4	1	no	—
63	C₂×C₁₆×C₇₉	—	4	1	no	—
64	C₂×C₁₆×C₇₉	C₂³	1	1	yes	characteristic
65	C₂×C₁₆×C₇₉	C₂³	1	1	yes	characteristic
66	#107×C₇₉	C₂³	1	2	yes	—
67	#103×C₇₉	—	2	2	no	—
68	C₂×Dic₁₅₈	—	4	2	no	—
69	C₂×D₃₁₆	—	2	2	no	—
70	C₂×C₂²⋊D₇₉	—	4	2	no	—
71	C₂×C₄×D₇₉	—	4	2	no	—
72	#11929611	—	4	4	no	—
73	D₆₃₂	—	2	2	no	—
74	D₆₃₂	—	2	2	no	—
75	#11929608	—	4	1	no	—
76	C₂×C₇₉⋊C₈	C₂×D₄	1	2	yes	—
77	C₄⋊Dic₇₉	—	2	2	no	—
78	C₄⋊Dic₇₉	—	4	2	no	—
79	C₄⋊Dic₇₉	D₈	1	1	yes	characteristic
80	C₄⋊Dic₇₉	D₈	1	1	yes	characteristic
81	Dic₇₉⋊C₄	—	4	2	no	—
82	C₂²⋊Dic₇₉	D₈	1	2	yes	—
83	(C₂×D₇₉)⋊C₄	—	4	2	no	—
84	(C₂×D₇₉)⋊C₄	—	4	2	no	—
85	(C₂×D₇₉)⋊C₄	—	4	2	no	—
86	(C₂×D₇₉)⋊C₄	—	4	2	no	—
87	C₄×Dic₇₉	—	4	2	no	—
88	C₄×Dic₇₉	C₂×D₄	1	2	yes	—
89	C₇₉⋊C₁₆	—	2	2	no	—
90	C₂×Q₈×C₇₉	—	2	2	no	—
91	C₂²×C₄×C₇₉	C₂×D₄	1	1	yes	characteristic
92	Q₁₆×C₇₉	—	2	2	no	—
93	Q₁₆×C₇₉	—	2	2	no	—
94	C₂×C₈×C₇₉	—	2	1	no	—
95	C₂×C₈×C₇₉	—	2	1	no	—
96	C₂×C₈×C₇₉	C₂×D₄	1	1	yes	characteristic
97	C₂×C₈×C₇₉	C₂⁴	1	1	yes	characteristic
98	C₄⋊C₄×C₇₉	—	2	2	no	—
99	C₄⋊C₄×C₇₉	—	4	2	no	—
100	C₂²⋊C₄×C₇₉	—	4	2	no	—
101	C₁₆×C₇₉	—	2	1	no	—
102	C₁₆×C₇₉	—	2	1	no	—
103	C₂³×C₇₉	C₂×D₈	1	1	yes	characteristic
104	C₂²×D₇₉	—	4	2	no	—
105	Q₈×C₇₉	—	4	2	no	—
106	Q₈×C₇₉	—	4	2	no	—
107	C₂×C₄×C₇₉	—	2	1	no	—
108	C₂×C₄×C₇₉	—	2	1	no	—
109	C₂×C₄×C₇₉	—	4	2	no	—
110	C₂×C₄×C₇₉	—	4	2	no	—
111	C₂×C₄×C₇₉	C₂×D₈	1	1	yes	characteristic
112	C₂×C₄×C₇₉	C₂²×D₄	1	1	yes	characteristic
113	C₂²⋊D₇₉	—	16	2	no	—
114	C₂×Dic₇₉	—	4	2	no	—
115	C₂×Dic₇₉	C₂×D₈	1	2	yes	—
116	C₂×Dic₇₉	C₂×D₈	1	2	yes	—
117	D₃₁₆	—	4	2	no	—
118	D₃₁₆	—	4	2	no	—
119	C₄×D₇₉	—	16	2	no	—
120	Dic₁₅₈	—	16	2	no	—
121	C₈×C₇₉	—	2	1	no	—
122	C₈×C₇₉	H(5,𝔽₂)	1	1	yes	characteristic
123	C₈×C₇₉	2₋¹⁺⁴	1	1	yes	characteristic
124	C₇₉⋊C₈	—	2	2	no	—
125	C₂²×C₇₉	—	2	1	no	—
126	C₂²×C₇₉	—	2	1	no	—
127	C₂²×C₇₉	—	2	1	no	—
128	C₂²×C₇₉	C₂²×D₈	1	1	yes	characteristic
129	C₂×D₇₉	—	8	2	no	—
130	C₂×D₇₉	—	8	2	no	—
131	C₂×D₇₉	—	8	2	no	—
132	C₄×C₇₉	—	2	1	no	—
133	C₄×C₇₉	—	8	2	no	—
134	C₄×C₇₉	—	8	2	no	—
135	C₄×C₇₉	#537	1	1	yes	characteristic
136	C₄×C₇₉	#575	1	1	yes	characteristic
137	Dic₇₉	—	2	2	no	—
138	Dic₇₉	—	2	2	no	—
139	Dic₇₉	—	8	2	no	—
140	#33828	—	79	1	no	maximal, Sylow
141	C₂×C₇₉	—	4	1	no	—
142	C₂×C₇₉	#3245	1	1	yes	characteristic
143	C₂×C₇₉	#3415	1	1	yes	characteristic
144	C₂×C₇₉	#3417	1	1	yes	characteristic
145	D₇₉	—	16	2	no	—
146	#3175	—	79	2	no	—
147	#2227	—	79	2	no	—
148	#2224	—	79	2	no	—
149	#2220	—	79	2	no	—
150	#2216	D₇₉	1	1	yes	characteristic, complemented
151	#2215	—	79	1	no	—
152	#2211	—	79	2	no	—
153	#2210	—	79	2	no	—
154	#2151	—	79	1	no	—
155	C₇₉	#33828	1	1	yes	characteristic, complemented, Sylow
156	#542	—	158	2	no	—
157	C₂×Q₃₂	C₂×D₇₉	1	1	yes	characteristic
158	C₂×SD₃₂	—	79	2	no	—
159	C₂×D₁₆	—	79	1	no	—
160	C₂×M₃₂	—	79	1	no	—
161	C₂²×C₁₆	C₂×D₇₉	1	1	yes	characteristic
162	#500	—	79	2	no	—
163	#495	—	79	2	no	—
164	#489	—	158	2	no	—
165	#468	—	158	2	no	—
166	#467	C₂×D₇₉	1	2	yes	—
167	#466	—	79	2	no	—
168	#440	—	158	2	no	—
169	C₄×Q₁₆	—	79	2	no	—
170	C₄×D₈	—	79	2	no	—
171	#405	—	79	1	no	—
172	#366	—	79	1	no	—
173	#366	—	79	2	no	—
174	#366	C₂×D₇₉	1	1	yes	characteristic
175	C₄⋊C₁₆	—	79	2	no	—
176	C₄⋊C₁₆	—	79	2	no	—
177	#358	—	79	2	no	—
178	#358	C₂×D₇₉	1	2	yes	—
179	#357	—	79	2	no	—
180	#357	—	79	2	no	—
181	C₂×Q₁₆	C₂²×D₇₉	1	2	yes	—
182	C₂×SD₁₆	—	158	2	no	—
183	C₂×D₈	—	79	2	no	—
184	C₂×M₁₆	—	79	1	no	—
185	C₂²×C₈	C₂²×D₇₉	1	1	yes	characteristic
186	C₄⋊Q₈	—	158	2	no	—
187	#126	—	316	2	no	—
188	#124	—	316	2	no	—
189	#124	—	158	2	no	—
190	#123	—	316	2	no	—
191	C₂²⋊Q₈	—	2	2	no	—
192	C₂²⋊Q₈	—	316	2	no	—
193	#121	—	158	2	no	—
194	C₄×Q₈	—	158	2	no	—
195	C₄×D₄	—	158	2	no	—
196	#117	—	79	1	no	—
197	Q₃₂	—	4	1	no	—
198	SD₃₂	—	316	2	no	—
199	D₁₆	—	316	1	no	—
200	M₃₂	—	316	1	no	—
201	C₂×C₁₆	—	4	1	no	—
202	C₂×C₁₆	—	79	2	no	—
203	C₂×C₁₆	C₂²×D₇₉	1	1	yes	characteristic
204	C₂×C₁₆	C₂²×D₇₉	1	1	yes	characteristic
205	#107	—	79	2	no	—
206	#107	—	158	2	no	—
207	#107	C₂²×D₇₉	1	2	yes	—
208	#103	—	2	2	no	—
209	#103	—	158	2	no	—
210	#102	—	158	2	no	—
211	#102	—	158	2	no	—
212	#97	—	79	2	no	—
213	C₄×C₈	—	79	2	no	—
214	C₂×Q₈	—	2	2	no	—
215	C₂×Q₈	—	316	2	no	—
216	C₂×D₄	—	316	2	no	—
217	C₂×D₄	—	158	2	no	—
218	C₂²×C₄	—	316	2	no	—
219	C₂²×C₄	D₄×D₇₉	1	1	yes	characteristic
220	Q₁₆	—	2	2	no	—
221	Q₁₆	—	2	2	no	—
222	SD₁₆	—	316	4	no	—
223	D₈	—	158	2	no	—
224	D₈	—	158	2	no	—
225	M₁₆	—	316	1	no	—
226	C₂×C₈	—	2	1	no	—
227	C₂×C₈	—	2	1	no	—
228	C₂×C₈	—	79	2	no	—
229	C₂×C₈	D₄×D₇₉	1	1	yes	characteristic
230	C₂×C₈	C₂³×D₇₉	1	1	yes	characteristic
231	C₄⋊C₄	—	2	2	no	—
232	C₄⋊C₄	—	4	2	no	—
233	C₄⋊C₄	—	316	2	no	—
234	C₄⋊C₄	—	316	2	no	—
235	C₄⋊C₄	—	79	1	no	—
236	C₄⋊C₄	—	79	1	no	—
237	C₄⋊C₄	—	158	2	no	—
238	C₂²⋊C₄	—	4	2	no	—
239	C₂²⋊C₄	—	316	2	no	—
240	C₂²⋊C₄	—	316	2	no	—
241	C₂²⋊C₄	—	316	2	no	—
242	C₂²⋊C₄	—	316	2	no	—
243	C₂²⋊C₄	—	79	2	no	—
244	C₄²	—	316	2	no	—
245	C₄²	—	79	2	no	—
246	C₁₆	—	2	1	no	—
247	C₁₆	—	2	1	no	—
248	C₁₆	—	158	2	no	—
249	C₂³	—	316	2	no	—
250	C₂³	D₈×D₇₉	1	1	yes	characteristic
251	Q₈	—	4	2	no	—
252	Q₈	—	4	2	no	—
253	Q₈	—	1264	2	no	—
254	D₄	—	316	2	no	—
255	D₄	—	316	2	no	—
256	D₄	—	1264	2	no	—
257	C₂×C₄	—	2	1	no	—
258	C₂×C₄	—	2	1	no	—
259	C₂×C₄	—	4	2	no	—
260	C₂×C₄	—	4	2	no	—
261	C₂×C₄	—	316	2	no	—
262	C₂×C₄	—	79	2	no	—
263	C₂×C₄	—	79	2	no	—
264	C₂×C₄	—	1264	2	no	—
265	C₂×C₄	D₈×D₇₉	1	1	yes	characteristic
266	C₂×C₄	C₂×D₄×D₇₉	1	1	yes	characteristic
267	C₈	—	2	1	no	—
268	C₈	—	158	2	no	—
269	C₈	#14579206	1	1	yes	characteristic
270	C₈	#14579208	1	1	yes	characteristic
271	C₂²	—	2	1	no	—
272	C₂²	—	2	1	no	—
273	C₂²	—	2	1	no	—
274	C₂²	—	632	2	no	—
275	C₂²	—	632	2	no	—
276	C₂²	—	632	2	no	—
277	C₂²	C₂×D₈×D₇₉	1	1	yes	characteristic
278	C₄	—	2	1	no	—
279	C₄	—	8	2	no	—
280	C₄	—	8	2	no	—
281	C₄	—	632	2	no	—
282	C₄	—	158	2	no	—
283	C₄	—	158	2	no	—
284	C₄	#19099140	1	1	yes	characteristic
285	C₄	#19099360	1	1	yes	characteristic
286	C₂	—	4	1	no	—
287	C₂	—	1264	2	no	—
288	C₂	#23774706	1	1	yes	characteristic
289	C₂	#23776538	1	1	yes	characteristic
290	C₂	#23776549	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#23774040

C₂

yes

maximal, complemented

#23765550

C₂

yes

maximal, complemented

#23765536

C₂

yes

maximal, complemented

#23765515

C₂

yes

maximal, complemented

#23765486

C₂

yes

maximal, characteristic

#23765464

C₂

yes

maximal, complemented

#23765458

C₂

yes

maximal, complemented

#23765158

C₂

yes

maximal, characteristic, complemented

C₇₉×#2216

C₂

yes

maximal, characteristic, complemented

#19099186

—

C₂×#14579104

C₂²

yes

—

C₂×D₁₂₆₄

C₂²

yes

characteristic

C₂×#14579101

C₂²

yes

characteristic

#19098967

C₂²

yes

—

#19098948

C₂²

yes

—

#19098924

—

#19098811

—

#19098799

C₂²

yes

—

#19098666

—

Q₁₆×Dic₇₉

C₂²

yes

—

#19098645

C₂²

yes

—

#19098518

C₂²

yes

characteristic

#19098389

C₂²

yes

characteristic

#19098388

C₂²

yes

—

Dic₇₉⋊C₁₆

C₂²

yes

—

Dic₇₉⋊C₁₆

C₂²

yes

—

#19098367

C₂²

yes

—

#19098365

C₂²

yes

—

#19098363

C₂²

yes

—

C₂×Q₃₂×C₇₉

C₂²

yes

characteristic

C₂²×C₁₆×C₇₉

C₂²

yes

characteristic

#467×C₇₉

C₂²

yes

—

#366×C₇₉

C₂²

yes

characteristic

#358×C₇₉

C₂²

yes

—

C₂×#11929611

—

C₂×D₆₃₂

C₂³

yes

—

C₂×#11929608

D₄

yes

—

—

—

—

—

—

—

Q₈×Dic₇₉

—

#14579123

—

#14579120

D₄

yes

characteristic

#14579104

—

D₁₂₆₄

—

#14579101

—

C₂×C₇₉⋊C₁₆

C₂³

yes

—

#14579095

C₂³

yes

—

—

—

—

—

D₄

yes

—

C₈×Dic₇₉

C₂³

yes

—

C₂×Q₁₆×C₇₉

C₂³

yes

—

C₂²×C₈×C₇₉

C₂³

yes

characteristic

C₂²⋊Q₈×C₇₉

—

Q₃₂×C₇₉

—

C₂×C₁₆×C₇₉

—

C₂×C₁₆×C₇₉

C₂³

yes

characteristic

C₂×C₁₆×C₇₉

C₂³

yes

characteristic

#107×C₇₉

C₂³

yes

—

#103×C₇₉

—

C₂×Dic₁₅₈

—

C₂×D₃₁₆

—

C₂×C₂²⋊D₇₉

—

C₂×C₄×D₇₉

—

#11929611

—

D₆₃₂

—

D₆₃₂

—

#11929608

—

C₂×C₇₉⋊C₈

C₂×D₄

yes

—

C₄⋊Dic₇₉

—

C₄⋊Dic₇₉

—

C₄⋊Dic₇₉

D₈

yes

characteristic

C₄⋊Dic₇₉

D₈

yes

characteristic

Dic₇₉⋊C₄

—

C₂²⋊Dic₇₉

D₈

yes

—

(C₂×D₇₉)⋊C₄

—

(C₂×D₇₉)⋊C₄

—

(C₂×D₇₉)⋊C₄

—

(C₂×D₇₉)⋊C₄

—

C₄×Dic₇₉

—

C₄×Dic₇₉

C₂×D₄

yes

—

C₇₉⋊C₁₆

—

C₂×Q₈×C₇₉

—

C₂²×C₄×C₇₉

C₂×D₄

yes

characteristic

Q₁₆×C₇₉

—

Q₁₆×C₇₉

—

C₂×C₈×C₇₉

—

C₂×C₈×C₇₉

—

C₂×C₈×C₇₉

C₂×D₄

yes

characteristic

C₂×C₈×C₇₉

C₂⁴

yes

characteristic

C₄⋊C₄×C₇₉

—

C₄⋊C₄×C₇₉

—

100

C₂²⋊C₄×C₇₉

—

101

C₁₆×C₇₉

—

102

C₁₆×C₇₉

—

103

C₂³×C₇₉

C₂×D₈

yes

characteristic

104

C₂²×D₇₉

—

105

Q₈×C₇₉

—

106

Q₈×C₇₉

—

107

C₂×C₄×C₇₉

—

108

C₂×C₄×C₇₉

—

109

C₂×C₄×C₇₉

—

110

C₂×C₄×C₇₉

—

111

C₂×C₄×C₇₉

C₂×D₈

yes

characteristic

112

C₂×C₄×C₇₉

C₂²×D₄

yes

characteristic

113

C₂²⋊D₇₉

—

114

C₂×Dic₇₉

—

115

C₂×Dic₇₉

C₂×D₈

yes

—

116

C₂×Dic₇₉

C₂×D₈

yes

—

117

D₃₁₆

—

118

D₃₁₆

—

119

C₄×D₇₉

—

120

Dic₁₅₈

—

121

C₈×C₇₉

—

122

C₈×C₇₉

H(5,𝔽₂)

yes

characteristic

123

C₈×C₇₉

2₋¹⁺⁴

yes

characteristic

124

C₇₉⋊C₈

—

125

C₂²×C₇₉

—

126

C₂²×C₇₉

—

127

C₂²×C₇₉

—

128

C₂²×C₇₉

C₂²×D₈

yes

characteristic

129

C₂×D₇₉

—

130

C₂×D₇₉

—

131

C₂×D₇₉

—

132

C₄×C₇₉

—

133

C₄×C₇₉

—

134

C₄×C₇₉

—

135

C₄×C₇₉

#537

yes

characteristic

136

C₄×C₇₉

#575

yes

characteristic

137

Dic₇₉

—

138

Dic₇₉

—

139

Dic₇₉

—

140

#33828

—

maximal, Sylow

141

C₂×C₇₉

—

142

C₂×C₇₉

#3245

yes

characteristic

143

C₂×C₇₉

#3415

yes

characteristic

144

C₂×C₇₉

#3417

yes

characteristic

145

D₇₉

—

146

#3175

—

147

#2227

—

148

#2224

—

149

#2220

—

150

#2216

D₇₉

yes

characteristic, complemented

151

#2215

—

152

#2211

—

153

#2210

—

154

#2151

—

155

C₇₉

#33828

yes

characteristic, complemented, Sylow

156

#542

—

158

—

157

C₂×Q₃₂

C₂×D₇₉

yes

characteristic

158

C₂×SD₃₂

—

159

C₂×D₁₆

—

160

C₂×M₃₂

—

161

C₂²×C₁₆

C₂×D₇₉

yes

characteristic

162

#500

—

163

#495

—

164

#489

—

158

—

165

#468

—

158

—

166

#467

C₂×D₇₉

yes

—

167

#466

—

168

#440

—

158

—

169

C₄×Q₁₆

—

170

C₄×D₈

—

171

#405

—

172

#366

—

173

#366

—

174

#366

C₂×D₇₉

yes

characteristic

175

C₄⋊C₁₆

—

176

C₄⋊C₁₆

—

177

#358

—

178

#358

C₂×D₇₉

yes

—

179

#357

—

180

#357

—

181

C₂×Q₁₆

C₂²×D₇₉

yes

—

182

C₂×SD₁₆

—

158

—

183

C₂×D₈

—

184

C₂×M₁₆

—

185

C₂²×C₈

C₂²×D₇₉

yes

characteristic

186

C₄⋊Q₈

—

158

—

187

#126

—

316

—

188

#124

—

316

—

189

#124

—

158

—

190

#123

—

316

—

191

C₂²⋊Q₈

—

192

C₂²⋊Q₈

—

316

—

193

#121

—

158

—

194

C₄×Q₈

—

158

—

195

C₄×D₄

—

158

—

196

#117

—

197

Q₃₂

—

198

SD₃₂

—

316

—

199

D₁₆

—

316

—

200

M₃₂

—

316

—

201

C₂×C₁₆

—

202

C₂×C₁₆

—

203

C₂×C₁₆

C₂²×D₇₉

yes

characteristic

204

C₂×C₁₆

C₂²×D₇₉

yes

characteristic

205

#107

—

206

#107

—

158

—

207

#107

C₂²×D₇₉

yes

—

208

#103

—

209

#103

—

158

—

210

#102

—

158

—

211

#102

—

158

—

212

#97

—

213

C₄×C₈

—

214

C₂×Q₈

—

215

C₂×Q₈

—

316

—

216

C₂×D₄

—

316

—

217

C₂×D₄

—

158

—

218

C₂²×C₄

—

316

—

219

C₂²×C₄

D₄×D₇₉

yes

characteristic

220

Q₁₆

—

221

Q₁₆

—

222

SD₁₆

—

316

—

223

D₈

—

158

—

224

D₈

—

158

—

225

M₁₆

—

316

—

226

C₂×C₈

—

227

C₂×C₈

—

228

C₂×C₈

—

229

C₂×C₈

D₄×D₇₉

yes

characteristic

230

C₂×C₈

C₂³×D₇₉

yes

characteristic

231

C₄⋊C₄

—

232

C₄⋊C₄

—

233

C₄⋊C₄

—

316

—

234

C₄⋊C₄

—

316

—

235

C₄⋊C₄

—

236

C₄⋊C₄

—

237

C₄⋊C₄

—

158

—

238

C₂²⋊C₄

—

239

C₂²⋊C₄

—

316

—

240

C₂²⋊C₄

—

316

—

241

C₂²⋊C₄

—

316

—

242

C₂²⋊C₄

—

316

—

243

C₂²⋊C₄

—

244

C₄²

—

316

—

245

C₄²

—

246

C₁₆

—

247

C₁₆

—

248

C₁₆

—

158

—

249

C₂³

—

316

—

250

C₂³

D₈×D₇₉

yes

characteristic

251

Q₈

—

252

Q₈

—

253

Q₈

—

1264

—

254

D₄

—

316

—

255

D₄

—

316

—

256

D₄

—

1264

—

257

C₂×C₄

—

258

C₂×C₄

—

259

C₂×C₄

—

260

C₂×C₄

—

261

C₂×C₄

—

316

—

262

C₂×C₄

—

263

C₂×C₄

—

264

C₂×C₄

—

1264

—

265

C₂×C₄

D₈×D₇₉

yes

characteristic

266

C₂×C₄

C₂×D₄×D₇₉

yes

characteristic

267

C₈

—

268

C₈

—

158

—

269

C₈

#14579206

yes

characteristic

270

C₈

#14579208

yes

characteristic

271

C₂²

—

272

C₂²

—

273

C₂²

—

274

C₂²

—

632

—

275

C₂²

—

632

—

276

C₂²

—

632

—

277

C₂²

C₂×D₈×D₇₉

yes

characteristic

278

C₄

—

279

C₄

—

280

C₄

—

281

C₄

—

632

—

282

C₄

—

158

—

283

C₄

—

158

—

284

C₄

#19099140

yes

characteristic

285

C₄

#19099360

yes

characteristic

286

C₂

—

287

C₂

—

1264

—

288

C₂

#23774706

yes

characteristic

289

C₂

#23776538

yes

characteristic

290

C₂

#23776549

yes

characteristic

	subgroup	G/H
center	C₂²	C₂×D₈×D₇₉
commutator, cosocle	C₂×C₈×C₇₉	C₂⁴
Frattini	C₂×C₈	C₂³×D₇₉
Fitting	C₇₉×#2216	C₂
socle	C₂²×C₇₉	C₂²×D₈
prime radical	#23765486	C₂
hypercenter	#2216	D₇₉
hypocenter	C₇₉	#33828
LCS	G ▷ C₂×C₈×C₇₉ ▷ C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ #2216

subgroup

G/H

center

C₂²

C₂×D₈×D₇₉

commutator, cosocle

C₂×C₈×C₇₉

C₂⁴

Frattini

C₂×C₈

C₂³×D₇₉

Fitting

C₇₉×#2216

C₂

socle

C₂²×C₇₉

C₂²×D₈

C₂

D₇₉

C₇₉

G ▷ C₂×C₈×C₇₉ ▷ C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ #2216

N	H ≅ G/N	type
1. #23774040	C₂	semidirect
2. #23765550	C₂	semidirect
3. #23765536	C₂	semidirect
4. #23765515	C₂	semidirect
6. #23765464	C₂	semidirect
7. #23765458	C₂	semidirect
8. #23765158	C₂	semidirect
9. C₇₉×#2216	C₂	semidirect
150. #2216	D₇₉	semidirect
155. C₇₉	#33828	semidirect

H ≅ G/N

type

1. #23774040

C₂

semidirect

2. #23765550

C₂

semidirect

3. #23765536

C₂

semidirect

4. #23765515

C₂

semidirect

6. #23765464

C₂

semidirect

7. #23765458

C₂

semidirect

8. #23765158

C₂

semidirect

9. C₇₉×#2216

C₂

semidirect

150. #2216

D₇₉

semidirect

155. C₇₉

#33828

semidirect

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

subgroup	index
#23774040	2
#23765550	2
#23765536	2
#23765515	2
#23765486	2
#23765464	2
#23765458	2
#23765158	2
C₇₉×#2216	2
#33828	79

subgroup

index

C₇₉×#2216

Subgroups — #34444709

Conjugacy classes of subgroups

Complements

Maximal subgroups