Subgroups — C₂×#22624898 #34010417

Order 18688.

382 conjugacy classes of subgroups.

The subgroup lattice has 240 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄⋊#11924031	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄×#11924031	C₂	1	1	yes	maximal, characteristic
3	#22624898	C₂	1	4	yes	maximal, direct product factor
4	C₂×#17988711	C₂	1	1	yes	maximal, characteristic
5	C₂²×#11924031	—	2	1	no	—
6	C₂²×#11924031	—	2	1	no	—
7	C₄⋊#11924031	—	2	1	no	—
8	C₄⋊#11924031	C₂²	1	2	yes	—
9	C₄×#11924031	—	2	1	no	—
10	C₄×#11924031	C₂²	1	2	yes	—
11	C₂×C₄×Dic₁₄₆	C₄	1	1	yes	characteristic
12	#17988711	—	2	1	no	—
13	#17988711	—	2	1	no	—
14	#17988711	C₄	1	2	yes	—
15	#17988711	C₂²	1	2	yes	—
16	C₂×#14556520	—	2	1	no	—
17	C₂×C₄×#10596831	C₂²	1	1	yes	characteristic
18	C₂×C₈×Dic₇₃	—	2	1	no	—
19	C₂×C₄⋊C₈×C₇₃	C₄	1	1	yes	characteristic
20	C₂×#11924031	—	2	4	no	—
21	C₂×#11924031	—	2	2	no	—
22	C₂×#11924031	—	2	2	no	—
23	C₂×#11924031	—	4	2	no	—
24	C₂²×Dic₁₄₆	C₈	1	1	yes	characteristic
25	C₂²×#10596831	D₄	1	1	yes	characteristic
26	C₂²×#10596831	D₄	1	1	yes	characteristic
27	C₄×Dic₁₄₆	—	2	1	no	—
28	C₄×Dic₁₄₆	—	2	1	no	—
29	C₄×Dic₁₄₆	C₈	1	2	yes	—
30	C₄×Dic₁₄₆	C₂×C₄	1	2	yes	—
31	C₂×C₄⋊Dic₇₃	C₈	1	1	yes	characteristic
32	C₂×#11924007	—	2	1	no	—
33	C₂×C₄×Dic₇₃	—	2	1	no	—
34	#14556520	—	2	2	no	—
35	#14556520	—	2	2	no	—
36	C₄×#10596831	C₂×C₄	1	1	yes	characteristic
37	C₄×#10596831	D₄	1	1	yes	characteristic
38	C₄×#10596831	D₄	1	1	yes	characteristic
39	C₄×#10596831	C₂³	1	1	yes	characteristic
40	C₈×Dic₇₃	—	2	2	no	—
41	C₈×Dic₇₃	—	2	2	no	—
42	C₂²×C₈×C₇₃	—	2	1	no	—
43	C₂×C₄²×C₇₃	C₂×C₄	1	1	yes	characteristic
44	C₄⋊C₈×C₇₃	—	2	1	no	—
45	C₄⋊C₈×C₇₃	C₂×C₄	1	2	yes	—
46	#11924031	—	2	4	no	—
47	#11924031	—	4	2	no	—
48	C₂×Dic₁₄₆	—	2	1	no	—
49	C₂×Dic₁₄₆	—	2	2	no	—
50	C₂×Dic₁₄₆	C₁₆	1	4	yes	complemented
51	C₂×Dic₁₄₆	C₂×C₈	1	2	yes	—
52	C₂²×Dic₇₃	—	2	1	no	—
53	C₂²×Dic₇₃	—	2	1	no	—
54	C₂×#10596831	—	2	2	no	—
55	C₂×#10596831	C₂²⋊C₄	1	1	yes	characteristic
56	C₂×#10596831	C₂²⋊C₄	1	1	yes	characteristic
57	C₂×#10596831	SD₁₆	1	2	yes	—
58	C₂×#10596831	Q₁₆	1	2	yes	—
59	C₂×#10596831	C₂×D₄	1	1	yes	characteristic
60	C₂×#10596831	C₂×D₄	1	1	yes	characteristic
61	C₄⋊Dic₇₃	—	2	1	no	—
62	C₄⋊Dic₇₃	C₂×C₈	1	2	yes	—
63	#11924007	—	2	2	no	—
64	#11924007	—	2	2	no	—
65	C₄×Dic₇₃	—	2	2	no	—
66	C₄×Dic₇₃	—	2	2	no	—
67	C₂²×C₄×C₇₃	C₂²⋊C₄	1	1	yes	characteristic
68	C₂²×C₄×C₇₃	C₂×C₈	1	1	yes	characteristic
69	C₂²×C₄×C₇₃	M₁₆	1	1	yes	characteristic
70	C₂×C₈×C₇₃	—	2	2	no	—
71	C₂×C₈×C₇₃	—	4	2	no	—
72	C₄²×C₇₃	C₂²⋊C₄	1	1	yes	characteristic
73	C₄²×C₇₃	C₂×C₈	1	1	yes	characteristic
74	C₄²×C₇₃	M₁₆	1	1	yes	characteristic
75	C₄²×C₇₃	C₂²×C₄	1	1	yes	characteristic
76	C₂³×C₇₃	C₂²⋊C₈	1	1	yes	characteristic
77	C₂×C₄×C₇₃	—	2	2	no	—
78	C₂×C₄×C₇₃	C₂²⋊C₈	1	1	yes	characteristic
79	C₂×C₄×C₇₃	C₂²⋊C₈	1	1	yes	characteristic
80	C₂×C₄×C₇₃	C₂²⋊C₈	1	1	yes	characteristic
81	C₂×C₄×C₇₃	#103	1	2	yes	—
82	C₂×C₄×C₇₃	#104	1	2	yes	—
83	C₂×C₄×C₇₃	C₂×C₁₆	1	2	yes	—
84	C₂×C₄×C₇₃	M₃₂	1	2	yes	—
85	C₂×C₄×C₇₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
86	C₂×C₄×C₇₃	C₂²×C₈	1	1	yes	characteristic
87	C₂×C₄×C₇₃	C₂×M₁₆	1	1	yes	characteristic
88	C₂×Dic₇₃	—	2	4	no	—
89	C₂×Dic₇₃	—	2	2	no	—
90	C₂×Dic₇₃	—	2	2	no	—
91	C₂×Dic₇₃	—	4	2	no	—
92	Dic₁₄₆	—	2	4	no	—
93	Dic₁₄₆	—	2	2	no	—
94	Dic₁₄₆	C₂×C₁₆	1	4	yes	complemented
95	C₈×C₇₃	—	4	2	no	—
96	#10596831	—	2	1	no	—
97	#10596831	—	2	1	no	—
98	#10596831	#103	1	2	yes	—
99	#10596831	C₂×SD₁₆	1	1	yes	characteristic
100	#10596831	C₂×Q₁₆	1	1	yes	characteristic
101	C₂²×C₇₃	Q₈⋊C₈	1	2	yes	—
102	C₂²×C₇₃	C₂²⋊C₁₆	1	2	yes	—
103	C₂²×C₇₃	#350	1	2	yes	—
104	C₂²×C₇₃	C₂×C₂²⋊C₈	1	1	yes	characteristic
105	C₄×C₇₃	—	2	1	no	—
106	C₄×C₇₃	—	2	1	no	—
107	C₄×C₇₃	Q₈⋊C₈	1	2	yes	—
108	C₄×C₇₃	C₂²⋊C₁₆	1	2	yes	—
109	C₄×C₇₃	C₂×#103	1	1	yes	characteristic
110	C₄×C₇₃	C₂×#104	1	1	yes	characteristic
111	C₄×C₇₃	C₂²×C₁₆	1	1	yes	characteristic
112	C₄×C₇₃	C₂×M₃₂	1	1	yes	characteristic
113	Dic₇₃	—	2	4	no	—
114	Dic₇₃	—	4	2	no	—
115	C₂×Q₈⋊C₁₆	—	73	1	no	maximal, Sylow
116	C₂×C₇₃	Q₈⋊C₁₆	1	4	yes	complemented
117	C₂×C₇₃	C₂×Q₈⋊C₈	1	1	yes	characteristic
118	C₂×C₇₃	C₂×C₂²⋊C₁₆	1	1	yes	characteristic
119	C₂×C₇₃	C₂×#350	1	1	yes	characteristic
120	C₂×C₈×Q₈	—	73	1	no	—
121	C₂×C₄⋊C₁₆	—	73	1	no	—
122	C₂×C₄×C₁₆	—	73	1	no	—
123	Q₈⋊C₁₆	—	73	4	no	—
124	C₇₃	C₂×Q₈⋊C₁₆	1	1	yes	characteristic, complemented, Sylow
125	C₂×C₄×Q₈	—	73	1	no	—
126	C₂²×C₁₆	—	146	1	no	—
127	C₂²×C₁₆	—	146	1	no	—
128	C₈×Q₈	—	73	2	no	—
129	C₈×Q₈	—	73	2	no	—
130	C₈×Q₈	—	146	1	no	—
131	C₈×Q₈	—	146	1	no	—
132	C₂×C₄⋊C₈	—	146	1	no	—
133	C₂×C₄⋊C₈	#64408	1	1	yes	characteristic
134	C₂×C₄×C₈	—	73	1	no	—
135	C₂×C₄×C₈	—	146	1	no	—
136	C₄⋊C₁₆	—	73	2	no	—
137	C₄⋊C₁₆	—	146	1	no	—
138	C₄×C₁₆	—	73	2	no	—
139	C₄×C₁₆	—	146	1	no	—
140	C₂²×Q₈	—	73	1	no	—
141	C₂²×C₈	—	2	1	no	—
142	C₂²×C₈	—	73	1	no	—
143	C₂²×C₈	—	73	1	no	—
144	C₄×Q₈	—	73	2	no	—
145	C₄×Q₈	—	73	2	no	—
146	C₄×Q₈	—	146	1	no	—
147	C₄×Q₈	—	146	1	no	—
148	C₂×C₄⋊C₄	—	73	1	no	—
149	C₂×C₄⋊C₄	—	146	1	no	—
150	C₂×C₄²	—	146	1	no	—
151	C₂×C₄²	C₂×#64408	1	1	yes	characteristic
152	C₂×C₁₆	—	292	2	no	—
153	C₂×C₁₆	—	146	2	no	—
154	C₂×C₁₆	—	146	2	no	—
155	C₂×C₁₆	—	146	4	no	—
156	C₄⋊C₈	—	2	1	no	—
157	C₄⋊C₈	—	146	2	no	—
158	C₄⋊C₈	—	146	2	no	—
159	C₄⋊C₈	C₂×#64408	1	2	yes	—
160	C₄×C₈	—	73	1	no	—
161	C₄×C₈	—	73	1	no	—
162	C₄×C₈	—	73	1	no	—
163	C₄×C₈	—	73	1	no	—
164	C₄×C₈	—	146	2	no	—
165	C₄×C₈	—	146	2	no	—
166	C₂×Q₈	—	73	2	no	—
167	C₂×Q₈	—	73	4	no	—
168	C₂×Q₈	—	146	1	no	—
169	C₂×Q₈	—	146	2	no	—
170	C₂²×C₄	—	146	1	no	—
171	C₂²×C₄	—	146	1	no	—
172	C₂²×C₄	C₂²⋊#64408	1	1	yes	characteristic
173	C₂²×C₄	#11924036	1	1	yes	characteristic
174	C₂²×C₄	#11924038	1	1	yes	characteristic
175	C₂×C₈	—	2	2	no	—
176	C₂×C₈	—	4	2	no	—
177	C₂×C₈	—	73	2	no	—
178	C₂×C₈	—	73	2	no	—
179	C₂×C₈	—	73	1	no	—
180	C₂×C₈	—	73	1	no	—
181	C₂×C₈	—	73	1	no	—
182	C₂×C₈	—	73	1	no	—
183	C₂×C₈	—	146	2	no	—
184	C₄⋊C₄	—	73	2	no	—
185	C₄⋊C₄	—	146	2	no	—
186	C₄⋊C₄	—	146	2	no	—
187	C₄⋊C₄	—	146	1	no	—
188	C₄²	—	146	2	no	—
189	C₄²	—	146	2	no	—
190	C₄²	C₂²⋊#64408	1	1	yes	characteristic
191	C₄²	#11924036	1	1	yes	characteristic
192	C₄²	#11924038	1	1	yes	characteristic
193	C₄²	C₂²×#64408	1	1	yes	characteristic
194	C₁₆	—	292	2	no	—
195	C₁₆	—	146	4	no	—
196	C₂³	#14556648	1	1	yes	characteristic
197	Q₈	—	73	4	no	—
198	Q₈	—	146	2	no	—
199	Q₈	—	146	4	no	—
200	C₂×C₄	—	2	2	no	—
201	C₂×C₄	—	292	2	no	—
202	C₂×C₄	—	146	2	no	—
203	C₂×C₄	—	146	4	no	—
204	C₂×C₄	—	146	2	no	—
205	C₂×C₄	#14556648	1	1	yes	characteristic
206	C₂×C₄	#14556648	1	1	yes	characteristic
207	C₂×C₄	#14556648	1	1	yes	characteristic
208	C₂×C₄	#14556654	1	2	yes	—
209	C₂×C₄	#14556655	1	2	yes	—
210	C₂×C₄	#14556664	1	2	yes	—
211	C₂×C₄	#14556666	1	2	yes	—
212	C₂×C₄	C₂×C₂²⋊#64408	1	1	yes	characteristic
213	C₂×C₄	C₂×#11924036	1	1	yes	characteristic
214	C₂×C₄	C₂×#11924038	1	1	yes	characteristic
215	C₈	—	4	2	no	—
216	C₈	—	73	1	no	—
217	C₈	—	73	2	no	—
218	C₈	—	73	1	no	—
219	C₈	—	146	1	no	—
220	C₈	—	146	1	no	—
221	C₂²	#17989437	1	2	yes	—
222	C₂²	#17989482	1	2	yes	—
223	C₂²	#17989486	1	2	yes	—
224	C₂²	C₂×#14556648	1	1	yes	characteristic
225	C₄	—	2	1	no	—
226	C₄	—	2	1	no	—
227	C₄	—	292	2	no	—
228	C₄	—	146	4	no	—
229	C₄	#17989437	1	2	yes	—
230	C₄	#17989482	1	2	yes	—
231	C₄	C₂×#14556654	1	1	yes	characteristic
232	C₄	C₂×#14556655	1	1	yes	characteristic
233	C₄	C₂×#14556664	1	1	yes	characteristic
234	C₄	C₂×#14556666	1	1	yes	characteristic
235	C₂	#22624898	1	4	yes	direct product factor
236	C₂	C₂×#17989437	1	1	yes	characteristic
237	C₂	C₂×#17989482	1	1	yes	characteristic
238	C₂	C₂×#17989486	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₂²⋊#64408
commutator	C₄×C₇₃	C₂²×C₁₆
Frattini	C₄²	C₂²×#64408
Fitting	C₂×C₄⋊C₈×C₇₃	C₄
socle, prime radical	C₂³×C₇₃	C₂²⋊C₈
cosocle	C₄×#10596831	C₂³
hypercenter	C₂×C₄⋊C₈	#64408
hypocenter	C₇₃	C₂×Q₈⋊C₁₆
LCS	G ▷ C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄² ◁ C₂×C₄⋊C₈

Complements

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

N	H ≅ G/N	type
3. #22624898	C₂	direct (N × H)
50. C₂×Dic₁₄₆	C₁₆	semidirect
94. Dic₁₄₆	C₂×C₁₆	semidirect
116. C₂×C₇₃	Q₈⋊C₁₆	semidirect
124. C₇₃	C₂×Q₈⋊C₁₆	semidirect
235. C₂	#22624898	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₄⋊#11924031	2
C₂×C₄×#11924031	2
#22624898	2
C₂×#17988711	2
C₂×Q₈⋊C₁₆	73

Subgroups — C2×#22624898 #34010417

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#22624898 #34010417