Subgroups — C₂×#1696×C₁₃₇ #46868535

Order 35072.

618 conjugacy classes of subgroups.

The subgroup lattice has 272 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#500×C₁₃₇	C₂	1	1	yes	maximal, characteristic
2	C₂×#477×C₁₃₇	C₂	1	2	yes	maximal
3	C₂×#446×C₁₃₇	C₂	1	1	yes	maximal, characteristic
4	#1696×C₁₃₇	C₂	1	8	yes	maximal, direct product factor
5	C₂×#334×C₁₃₇	C₂	1	1	yes	maximal, characteristic
6	C₂×#333×C₁₃₇	C₂	1	2	yes	maximal
7	C₂×C₄⋊Q₈×C₁₃₇	C₂²	1	2	yes	—
8	C₂×C₄×Q₈×C₁₃₇	C₂²	1	1	yes	characteristic
9	#500×C₁₃₇	—	2	2	no	—
10	#500×C₁₃₇	C₂²	1	4	yes	—
11	#477×C₁₃₇	—	2	4	no	—
12	#477×C₁₃₇	C₂²	1	8	yes	—
13	#446×C₁₃₇	—	2	2	no	—
14	#446×C₁₃₇	C₂²	1	4	yes	—
15	C₂×#107×C₁₃₇	—	2	1	no	—
16	C₂×#107×C₁₃₇	—	2	1	no	—
17	C₂×#107×C₁₃₇	—	2	2	no	—
18	C₂×C₄⋊C₈×C₁₃₇	—	2	1	no	—
19	C₂×C₄⋊C₈×C₁₃₇	C₂²	1	2	yes	—
20	C₂×C₄⋊C₈×C₁₃₇	C₂²	1	1	yes	characteristic
21	C₂×#103×C₁₃₇	—	2	2	no	—
22	C₂×#97×C₁₃₇	—	2	1	no	—
23	C₂×C₄×C₈×C₁₃₇	C₂²	1	1	yes	characteristic
24	#334×C₁₃₇	C₂²	1	4	yes	—
25	#333×C₁₃₇	C₂²	1	8	yes	—
26	C₂²×Q₈×C₁₃₇	—	2	2	no	—
27	C₂²×Q₈×C₁₃₇	D₄	1	1	yes	characteristic
28	C₂²×C₈×C₁₃₇	—	2	1	no	—
29	C₂²×C₈×C₁₃₇	—	2	2	no	—
30	C₂²×C₈×C₁₃₇	D₄	1	1	yes	characteristic
31	C₂²×C₈×C₁₃₇	D₄	1	1	yes	characteristic
32	C₄⋊Q₈×C₁₃₇	—	2	2	no	—
33	C₄⋊Q₈×C₁₃₇	—	2	2	no	—
34	C₄⋊Q₈×C₁₃₇	—	2	2	no	—
35	C₄⋊Q₈×C₁₃₇	C₂³	1	4	yes	—
36	C₄×Q₈×C₁₃₇	—	2	1	no	—
37	C₄×Q₈×C₁₃₇	—	2	1	no	—
38	C₄×Q₈×C₁₃₇	D₄	1	2	yes	—
39	C₄×Q₈×C₁₃₇	C₂³	1	2	yes	—
40	C₂×C₄⋊C₄×C₁₃₇	—	2	1	no	—
41	C₂×C₄⋊C₄×C₁₃₇	—	2	2	no	—
42	C₂×C₄⋊C₄×C₁₃₇	—	2	2	no	—
43	C₂×C₄⋊C₄×C₁₃₇	D₄	1	1	yes	characteristic
44	C₂×C₄²×C₁₃₇	—	2	1	no	—
45	C₂×C₄²×C₁₃₇	C₂³	1	1	yes	characteristic
46	#107×C₁₃₇	—	2	4	no	—
47	#107×C₁₃₇	—	2	4	no	—
48	#107×C₁₃₇	—	2	8	no	—
49	C₄⋊C₈×C₁₃₇	—	2	4	no	—
50	C₄⋊C₈×C₁₃₇	—	2	1	no	—
51	C₄⋊C₈×C₁₃₇	—	2	2	no	—
52	C₄⋊C₈×C₁₃₇	C₂³	1	2	yes	—
53	C₄⋊C₈×C₁₃₇	C₂³	1	4	yes	—
54	#103×C₁₃₇	—	2	8	no	—
55	#97×C₁₃₇	—	2	4	no	—
56	C₄×C₈×C₁₃₇	D₄	1	2	yes	—
57	C₄×C₈×C₁₃₇	C₂³	1	2	yes	—
58	C₂×Q₈×C₁₃₇	—	2	1	no	—
59	C₂×Q₈×C₁₃₇	—	2	4	no	—
60	C₂×Q₈×C₁₃₇	—	2	4	no	—
61	C₂×Q₈×C₁₃₇	—	4	2	no	—
62	C₂×Q₈×C₁₃₇	—	8	4	no	—
63	C₂×Q₈×C₁₃₇	C₂×D₄	1	2	yes	—
64	C₂²×C₄×C₁₃₇	—	2	1	no	—
65	C₂²×C₄×C₁₃₇	—	2	1	no	—
66	C₂²×C₄×C₁₃₇	—	4	2	no	—
67	C₂²×C₄×C₁₃₇	C₂×D₄	1	1	yes	characteristic
68	C₂²×C₄×C₁₃₇	C₂×D₄	1	1	yes	characteristic
69	C₂²×C₄×C₁₃₇	#41	1	1	yes	characteristic
70	C₂×C₈×C₁₃₇	—	2	2	no	—
71	C₂×C₈×C₁₃₇	—	2	2	no	—
72	C₂×C₈×C₁₃₇	—	2	4	no	—
73	C₂×C₈×C₁₃₇	—	4	2	no	—
74	C₂×C₈×C₁₃₇	—	4	4	no	—
75	C₂×C₈×C₁₃₇	Q₁₆	1	4	yes	—
76	C₂×C₈×C₁₃₇	C₂×D₄	1	2	yes	—
77	C₂×C₈×C₁₃₇	C₂×D₄	1	2	yes	—
78	C₄⋊C₄×C₁₃₇	—	2	2	no	—
79	C₄⋊C₄×C₁₃₇	—	2	1	no	—
80	C₄⋊C₄×C₁₃₇	—	2	2	no	—
81	C₄⋊C₄×C₁₃₇	—	2	4	no	—
82	C₄⋊C₄×C₁₃₇	—	2	4	no	—
83	C₄⋊C₄×C₁₃₇	—	4	2	no	—
84	C₄⋊C₄×C₁₃₇	—	4	2	no	—
85	C₄⋊C₄×C₁₃₇	C₂×D₄	1	2	yes	—
86	C₄²×C₁₃₇	—	2	2	no	—
87	C₄²×C₁₃₇	—	2	2	no	—
88	C₄²×C₁₃₇	C₂×D₄	1	1	yes	characteristic
89	C₄²×C₁₃₇	C₂×D₄	1	1	yes	characteristic
90	C₄²×C₁₃₇	#41	1	1	yes	characteristic
91	C₄²×C₁₃₇	C₂⁴	1	1	yes	characteristic
92	C₂³×C₁₃₇	#121	1	1	yes	characteristic
93	Q₈×C₁₃₇	—	2	4	no	—
94	Q₈×C₁₃₇	—	4	2	no	—
95	Q₈×C₁₃₇	—	8	4	no	—
96	Q₈×C₁₃₇	—	8	4	no	—
97	C₂×C₄×C₁₃₇	—	2	2	no	—
98	C₂×C₄×C₁₃₇	—	2	2	no	—
99	C₂×C₄×C₁₃₇	—	2	2	no	—
100	C₂×C₄×C₁₃₇	—	4	2	no	—
101	C₂×C₄×C₁₃₇	—	4	2	no	—
102	C₂×C₄×C₁₃₇	—	4	4	no	—
103	C₂×C₄×C₁₃₇	—	8	4	no	—
104	C₂×C₄×C₁₃₇	#121	1	1	yes	characteristic
105	C₂×C₄×C₁₃₇	#121	1	1	yes	characteristic
106	C₂×C₄×C₁₃₇	#121	1	1	yes	characteristic
107	C₂×C₄×C₁₃₇	C₂×Q₁₆	1	2	yes	—
108	C₂×C₄×C₁₃₇	#137	1	2	yes	—
109	C₂×C₄×C₁₃₇	#137	1	4	yes	—
110	C₂×C₄×C₁₃₇	C₂²×D₄	1	1	yes	characteristic
111	C₂×C₄×C₁₃₇	C₂²×D₄	1	1	yes	characteristic
112	C₂×C₄×C₁₃₇	C₂×#41	1	1	yes	characteristic
113	C₈×C₁₃₇	—	2	2	no	—
114	C₈×C₁₃₇	—	4	2	no	—
115	C₈×C₁₃₇	—	4	4	no	—
116	C₈×C₁₃₇	C₂×Q₁₆	1	4	yes	—
117	C₂²×C₁₃₇	#462	1	2	yes	—
118	C₂²×C₁₃₇	#470	1	2	yes	—
119	C₂²×C₁₃₇	#472	1	2	yes	—
120	C₂²×C₁₃₇	C₂×#121	1	1	yes	characteristic
121	C₄×C₁₃₇	—	2	1	no	—
122	C₄×C₁₃₇	—	2	1	no	—
123	C₄×C₁₃₇	—	4	2	no	—
124	C₄×C₁₃₇	—	4	2	no	—
125	C₄×C₁₃₇	—	8	4	no	—
126	C₄×C₁₃₇	#462	1	2	yes	—
127	C₄×C₁₃₇	#470	1	2	yes	—
128	C₄×C₁₃₇	C₂²×Q₁₆	1	1	yes	characteristic
129	C₄×C₁₃₇	C₂×#137	1	1	yes	characteristic
130	C₄×C₁₃₇	C₂×#137	1	2	yes	—
131	C₂×C₁₃₇	#1696	1	4	yes	direct product factor
132	C₂×C₁₃₇	C₂×#462	1	1	yes	characteristic
133	C₂×C₁₃₇	C₂×#470	1	1	yes	characteristic
134	C₂×C₁₃₇	C₂×#472	1	1	yes	characteristic
135	C₂×#1696	C₁₃₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
136	C₁₃₇	C₂×#1696	1	1	yes	characteristic, direct product factor, Sylow
137	C₂×#500	C₂×C₁₃₇	1	1	yes	characteristic
138	C₂×#477	C₂×C₁₃₇	1	2	yes	—
139	C₂×#446	C₂×C₁₃₇	1	1	yes	characteristic
140	#1696	C₂×C₁₃₇	1	8	yes	direct product factor
141	C₂×#334	C₂×C₁₃₇	1	1	yes	characteristic
142	C₂×#333	C₂×C₁₃₇	1	2	yes	—
143	C₂×C₄⋊Q₈	C₂²×C₁₃₇	1	2	yes	—
144	C₂×C₄×Q₈	C₂²×C₁₃₇	1	1	yes	characteristic
145	#500	—	2	2	no	—
146	#500	C₂²×C₁₃₇	1	4	yes	—
147	#477	—	2	4	no	—
148	#477	C₂²×C₁₃₇	1	8	yes	—
149	#446	—	2	2	no	—
150	#446	C₂²×C₁₃₇	1	4	yes	—
151	C₂×#107	—	2	1	no	—
152	C₂×#107	—	2	1	no	—
153	C₂×#107	—	2	2	no	—
154	C₂×C₄⋊C₈	—	2	1	no	—
155	C₂×C₄⋊C₈	C₂²×C₁₃₇	1	2	yes	—
156	C₂×C₄⋊C₈	C₂²×C₁₃₇	1	1	yes	characteristic
157	C₂×#103	—	2	2	no	—
158	C₂×#97	—	2	1	no	—
159	C₂×C₄×C₈	C₂²×C₁₃₇	1	1	yes	characteristic
160	#334	C₂²×C₁₃₇	1	4	yes	—
161	#333	C₂²×C₁₃₇	1	8	yes	—
162	C₂²×Q₈	—	2	2	no	—
163	C₂²×Q₈	D₄×C₁₃₇	1	1	yes	characteristic
164	C₂²×C₈	—	2	1	no	—
165	C₂²×C₈	—	2	2	no	—
166	C₂²×C₈	D₄×C₁₃₇	1	1	yes	characteristic
167	C₂²×C₈	D₄×C₁₃₇	1	1	yes	characteristic
168	C₄⋊Q₈	—	2	2	no	—
169	C₄⋊Q₈	—	2	2	no	—
170	C₄⋊Q₈	—	2	2	no	—
171	C₄⋊Q₈	C₂³×C₁₃₇	1	4	yes	—
172	C₄×Q₈	—	2	1	no	—
173	C₄×Q₈	—	2	1	no	—
174	C₄×Q₈	D₄×C₁₃₇	1	2	yes	—
175	C₄×Q₈	C₂³×C₁₃₇	1	2	yes	—
176	C₂×C₄⋊C₄	—	2	1	no	—
177	C₂×C₄⋊C₄	—	2	2	no	—
178	C₂×C₄⋊C₄	—	2	2	no	—
179	C₂×C₄⋊C₄	D₄×C₁₃₇	1	1	yes	characteristic
180	C₂×C₄²	—	2	1	no	—
181	C₂×C₄²	C₂³×C₁₃₇	1	1	yes	characteristic
182	#107	—	2	4	no	—
183	#107	—	2	4	no	—
184	#107	—	2	8	no	—
185	C₄⋊C₈	—	2	4	no	—
186	C₄⋊C₈	—	2	1	no	—
187	C₄⋊C₈	—	2	2	no	—
188	C₄⋊C₈	C₂³×C₁₃₇	1	2	yes	—
189	C₄⋊C₈	C₂³×C₁₃₇	1	4	yes	—
190	#103	—	2	8	no	—
191	#97	—	2	4	no	—
192	C₄×C₈	D₄×C₁₃₇	1	2	yes	—
193	C₄×C₈	C₂³×C₁₃₇	1	2	yes	—
194	C₂×Q₈	—	2	1	no	—
195	C₂×Q₈	—	2	4	no	—
196	C₂×Q₈	—	2	4	no	—
197	C₂×Q₈	—	4	2	no	—
198	C₂×Q₈	—	8	4	no	—
199	C₂×Q₈	C₂×D₄×C₁₃₇	1	2	yes	—
200	C₂²×C₄	—	2	1	no	—
201	C₂²×C₄	—	2	1	no	—
202	C₂²×C₄	—	4	2	no	—
203	C₂²×C₄	C₂×D₄×C₁₃₇	1	1	yes	characteristic
204	C₂²×C₄	C₂×D₄×C₁₃₇	1	1	yes	characteristic
205	C₂²×C₄	#41×C₁₃₇	1	1	yes	characteristic
206	C₂×C₈	—	2	2	no	—
207	C₂×C₈	—	2	2	no	—
208	C₂×C₈	—	2	4	no	—
209	C₂×C₈	—	4	2	no	—
210	C₂×C₈	—	4	4	no	—
211	C₂×C₈	Q₁₆×C₁₃₇	1	4	yes	—
212	C₂×C₈	C₂×D₄×C₁₃₇	1	2	yes	—
213	C₂×C₈	C₂×D₄×C₁₃₇	1	2	yes	—
214	C₄⋊C₄	—	2	2	no	—
215	C₄⋊C₄	—	2	1	no	—
216	C₄⋊C₄	—	2	2	no	—
217	C₄⋊C₄	—	2	4	no	—
218	C₄⋊C₄	—	2	4	no	—
219	C₄⋊C₄	—	4	2	no	—
220	C₄⋊C₄	—	4	2	no	—
221	C₄⋊C₄	C₂×D₄×C₁₃₇	1	2	yes	—
222	C₄²	—	2	2	no	—
223	C₄²	—	2	2	no	—
224	C₄²	C₂×D₄×C₁₃₇	1	1	yes	characteristic
225	C₄²	C₂×D₄×C₁₃₇	1	1	yes	characteristic
226	C₄²	#41×C₁₃₇	1	1	yes	characteristic
227	C₄²	C₂⁴×C₁₃₇	1	1	yes	characteristic
228	C₂³	#121×C₁₃₇	1	1	yes	characteristic
229	Q₈	—	2	4	no	—
230	Q₈	—	4	2	no	—
231	Q₈	—	8	4	no	—
232	Q₈	—	8	4	no	—
233	C₂×C₄	—	2	2	no	—
234	C₂×C₄	—	2	2	no	—
235	C₂×C₄	—	2	2	no	—
236	C₂×C₄	—	4	2	no	—
237	C₂×C₄	—	4	2	no	—
238	C₂×C₄	—	4	4	no	—
239	C₂×C₄	—	8	4	no	—
240	C₂×C₄	#121×C₁₃₇	1	1	yes	characteristic
241	C₂×C₄	#121×C₁₃₇	1	1	yes	characteristic
242	C₂×C₄	#121×C₁₃₇	1	1	yes	characteristic
243	C₂×C₄	C₂×Q₁₆×C₁₃₇	1	2	yes	—
244	C₂×C₄	#137×C₁₃₇	1	2	yes	—
245	C₂×C₄	#137×C₁₃₇	1	4	yes	—
246	C₂×C₄	C₂²×D₄×C₁₃₇	1	1	yes	characteristic
247	C₂×C₄	C₂²×D₄×C₁₃₇	1	1	yes	characteristic
248	C₂×C₄	C₂×#41×C₁₃₇	1	1	yes	characteristic
249	C₈	—	2	2	no	—
250	C₈	—	4	4	no	—
251	C₈	—	4	2	no	—
252	C₈	C₂×Q₁₆×C₁₃₇	1	4	yes	—
253	C₂²	#462×C₁₃₇	1	2	yes	—
254	C₂²	#470×C₁₃₇	1	2	yes	—
255	C₂²	#472×C₁₃₇	1	2	yes	—
256	C₂²	C₂×#121×C₁₃₇	1	1	yes	characteristic
257	C₄	—	2	1	no	—
258	C₄	—	2	1	no	—
259	C₄	—	4	2	no	—
260	C₄	—	4	2	no	—
261	C₄	—	8	4	no	—
262	C₄	#462×C₁₃₇	1	2	yes	—
263	C₄	#470×C₁₃₇	1	2	yes	—
264	C₄	C₂²×Q₁₆×C₁₃₇	1	1	yes	characteristic
265	C₄	C₂×#137×C₁₃₇	1	2	yes	—
266	C₄	C₂×#137×C₁₃₇	1	1	yes	characteristic
267	C₂	#1696×C₁₃₇	1	4	yes	direct product factor
268	C₂	C₂×#462×C₁₃₇	1	1	yes	characteristic
269	C₂	C₂×#470×C₁₃₇	1	1	yes	characteristic
270	C₂	C₂×#472×C₁₃₇	1	1	yes	characteristic

	subgroup	G/H
center, socle, prime radical	C₂³×C₁₃₇	#121
commutator, Frattini, cosocle	C₄²	C₂⁴×C₁₃₇
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³×C₁₃₇ ◁ C₂×C₄²×C₁₃₇ ◁ G

Complements

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

N	H ≅ G/N	type
4. #1696×C₁₃₇	C₂	direct (N × H)
131. C₂×C₁₃₇	#1696	direct (N × H)
135. C₂×#1696	C₁₃₇	direct (N × H)
136. C₁₃₇	C₂×#1696	direct (N × H)
140. #1696	C₂×C₁₃₇	direct (N × H)
267. C₂	#1696×C₁₃₇	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×#500×C₁₃₇	2
C₂×#477×C₁₃₇	2
C₂×#446×C₁₃₇	2
#1696×C₁₃₇	2
C₂×#334×C₁₃₇	2
C₂×#333×C₁₃₇	2
C₂×#1696	137

Subgroups — C2×#1696×C137 #46868535

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#1696×C₁₃₇ #46868535