Subgroups — C₂×C₆₇×#1558 #30696983

Order 17152.

1182 conjugacy classes of subgroups.

The subgroup lattice has 252 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₄⋊Q₈×C₆₇	C₂	1	1	yes	maximal, characteristic
2	C₂×#432×C₆₇	C₂	1	2	yes	maximal
3	C₆₇×#1558	C₂	1	8	yes	maximal, direct product factor
4	C₂×#333×C₆₇	C₂	1	4	yes	maximal, complemented
5	C₂³×Q₈×C₆₇	C₄	1	2	yes	—
6	C₂×C₄⋊Q₈×C₆₇	—	2	2	no	—
7	C₂×C₄⋊Q₈×C₆₇	—	2	4	no	—
8	C₂×C₄⋊Q₈×C₆₇	C₄	1	4	yes	—
9	C₂×C₄⋊Q₈×C₆₇	C₄	1	2	yes	—
10	C₂×C₄⋊Q₈×C₆₇	C₂²	1	2	yes	—
11	C₂×C₄⋊Q₈×C₆₇	C₂²	1	4	yes	—
12	C₂²×C₄⋊C₄×C₆₇	—	2	2	no	—
13	C₂²×C₄²×C₆₇	C₂²	1	1	yes	characteristic
14	#432×C₆₇	—	2	4	no	—
15	#432×C₆₇	C₂²	1	8	yes	—
16	C₂×C₄⋊C₈×C₆₇	C₂²	1	4	yes	—
17	C₂×C₂²⋊C₈×C₆₇	—	2	2	no	—
18	C₂×#97×C₆₇	—	2	2	no	—
19	#333×C₆₇	C₂²	1	16	yes	complemented
20	C₂²×Q₈×C₆₇	—	2	4	no	—
21	C₂²×Q₈×C₆₇	—	2	2	no	—
22	C₂²×Q₈×C₆₇	—	4	8	no	—
23	C₂²×Q₈×C₆₇	C₂×C₄	1	8	yes	—
24	C₂²×Q₈×C₆₇	C₂×C₄	1	4	yes	—
25	C₂³×C₄×C₆₇	—	2	2	no	—
26	C₂³×C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
27	C₂³×C₄×C₆₇	D₄	1	2	yes	—
28	C₂²×C₈×C₆₇	—	2	4	no	—
29	C₄⋊Q₈×C₆₇	—	2	4	no	—
30	C₄⋊Q₈×C₆₇	—	2	8	no	—
31	C₄⋊Q₈×C₆₇	—	2	4	no	—
32	C₄⋊Q₈×C₆₇	—	2	8	no	—
33	C₄⋊Q₈×C₆₇	C₂×C₄	1	4	yes	—
34	C₄⋊Q₈×C₆₇	C₂×C₄	1	8	yes	—
35	C₄⋊Q₈×C₆₇	C₂³	1	4	yes	—
36	C₂×C₄⋊C₄×C₆₇	—	2	8	no	—
37	C₂×C₄⋊C₄×C₆₇	—	2	4	no	—
38	C₂×C₄⋊C₄×C₆₇	—	2	8	no	—
39	C₂×C₄⋊C₄×C₆₇	—	2	4	no	—
40	C₂×C₄²×C₆₇	C₂×C₄	1	2	yes	—
41	C₂×C₄²×C₆₇	C₂×C₄	1	1	yes	characteristic
42	C₂×C₄²×C₆₇	D₄	1	2	yes	—
43	C₂×C₄²×C₆₇	D₄	1	4	yes	—
44	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
45	C₂×C₄²×C₆₇	C₂³	1	2	yes	—
46	C₄⋊C₈×C₆₇	—	2	4	no	—
47	C₄⋊C₈×C₆₇	C₂³	1	8	yes	—
48	C₂²⋊C₈×C₆₇	—	2	8	no	—
49	#97×C₆₇	—	2	8	no	—
50	C₂⁴×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
51	C₂×Q₈×C₆₇	—	2	4	no	—
52	C₂×Q₈×C₆₇	—	2	8	no	—
53	C₂×Q₈×C₆₇	—	4	16	no	—
54	C₂×Q₈×C₆₇	—	4	8	no	—
55	C₂×Q₈×C₆₇	—	4	16	no	—
56	C₂×Q₈×C₆₇	—	4	8	no	—
57	C₂×Q₈×C₆₇	C₂²×C₄	1	8	yes	—
58	C₂²×C₄×C₆₇	—	2	4	no	—
59	C₂²×C₄×C₆₇	—	2	8	no	—
60	C₂²×C₄×C₆₇	—	2	8	no	—
61	C₂²×C₄×C₆₇	—	2	4	no	—
62	C₂²×C₄×C₆₇	—	4	8	no	—
63	C₂²×C₄×C₆₇	C₂²⋊C₄	1	4	yes	—
64	C₂²×C₄×C₆₇	C₂²⋊C₄	1	2	yes	—
65	C₂²×C₄×C₆₇	C₂²⋊C₄	1	2	yes	—
66	C₂²×C₄×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
67	C₂²×C₄×C₆₇	C₂²×C₄	1	2	yes	—
68	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
69	C₂²×C₄×C₆₇	C₂×D₄	1	4	yes	—
70	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
71	C₂×C₈×C₆₇	—	2	8	no	—
72	C₂×C₈×C₆₇	—	4	8	no	—
73	C₄⋊C₄×C₆₇	—	2	16	no	—
74	C₄⋊C₄×C₆₇	—	2	8	no	—
75	C₄⋊C₄×C₆₇	—	2	8	no	—
76	C₄²×C₆₇	C₂²⋊C₄	1	2	yes	—
77	C₄²×C₆₇	C₂²⋊C₄	1	4	yes	—
78	C₄²×C₆₇	C₂²×C₄	1	2	yes	—
79	C₄²×C₆₇	C₂²×C₄	1	1	yes	characteristic
80	C₄²×C₆₇	C₂×D₄	1	2	yes	—
81	C₄²×C₆₇	C₂×D₄	1	4	yes	—
82	C₄²×C₆₇	C₂⁴	1	1	yes	characteristic
83	C₂³×C₆₇	—	2	4	no	—
84	C₂³×C₆₇	#100	1	4	yes	—
85	C₂³×C₆₇	C₂×C₂²⋊C₄	1	2	yes	—
86	C₂³×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
87	Q₈×C₆₇	—	4	8	no	—
88	Q₈×C₆₇	—	4	8	no	—
89	Q₈×C₆₇	—	4	16	no	—
90	C₂×C₄×C₆₇	—	2	4	no	—
91	C₂×C₄×C₆₇	—	2	2	no	—
92	C₂×C₄×C₆₇	—	2	4	no	—
93	C₂×C₄×C₆₇	—	2	2	no	—
94	C₂×C₄×C₆₇	—	2	4	no	—
95	C₂×C₄×C₆₇	—	2	8	no	—
96	C₂×C₄×C₆₇	—	2	8	no	—
97	C₂×C₄×C₆₇	—	2	8	no	—
98	C₂×C₄×C₆₇	—	4	16	no	—
99	C₂×C₄×C₆₇	—	4	8	no	—
100	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	4	yes	—
101	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
102	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	2	yes	—
103	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	2	yes	—
104	C₂×C₄×C₆₇	#137	1	8	yes	—
105	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
106	C₂×C₄×C₆₇	C₂²×D₄	1	2	yes	—
107	C₈×C₆₇	—	4	8	no	—
108	C₂²×C₆₇	—	2	8	no	—
109	C₂²×C₆₇	—	2	4	no	—
110	C₂²×C₆₇	C₂×#100	1	4	yes	—
111	C₂²×C₆₇	C₂×#100	1	2	yes	—
112	C₂²×C₆₇	#419	1	4	yes	—
113	C₂²×C₆₇	C₂²×C₂²⋊C₄	1	1	yes	characteristic
114	C₄×C₆₇	—	2	1	no	—
115	C₄×C₆₇	—	2	2	no	—
116	C₄×C₆₇	—	2	1	no	—
117	C₄×C₆₇	—	2	4	no	—
118	C₄×C₆₇	—	4	8	no	—
119	C₄×C₆₇	#419	1	4	yes	—
120	C₄×C₆₇	C₂×#137	1	4	yes	—
121	C₂×#1558	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
122	C₂×C₆₇	—	2	4	no	—
123	C₂×C₆₇	#1558	1	4	yes	direct product factor
124	C₂×C₆₇	C₂²×#100	1	1	yes	characteristic
125	C₂×C₆₇	C₂×#419	1	2	yes	—
126	C₂²×C₄⋊Q₈	C₂×C₆₇	1	1	yes	characteristic
127	C₂×#432	C₂×C₆₇	1	2	yes	—
128	#1558	C₂×C₆₇	1	8	yes	direct product factor
129	C₂×#333	C₂×C₆₇	1	4	yes	complemented
130	C₆₇	C₂×#1558	1	1	yes	characteristic, direct product factor, Sylow
131	C₂³×Q₈	C₄×C₆₇	1	2	yes	—
132	C₂×C₄⋊Q₈	—	2	2	no	—
133	C₂×C₄⋊Q₈	—	2	4	no	—
134	C₂×C₄⋊Q₈	C₄×C₆₇	1	4	yes	—
135	C₂×C₄⋊Q₈	C₄×C₆₇	1	2	yes	—
136	C₂×C₄⋊Q₈	C₂²×C₆₇	1	2	yes	—
137	C₂×C₄⋊Q₈	C₂²×C₆₇	1	4	yes	—
138	C₂²×C₄⋊C₄	—	2	2	no	—
139	C₂²×C₄²	C₂²×C₆₇	1	1	yes	characteristic
140	#432	—	2	4	no	—
141	#432	C₂²×C₆₇	1	8	yes	—
142	C₂×C₄⋊C₈	C₂²×C₆₇	1	4	yes	—
143	C₂×C₂²⋊C₈	—	2	2	no	—
144	C₂×#97	—	2	2	no	—
145	#333	C₂²×C₆₇	1	16	yes	complemented
146	C₂²×Q₈	—	2	4	no	—
147	C₂²×Q₈	—	2	2	no	—
148	C₂²×Q₈	—	4	8	no	—
149	C₂²×Q₈	C₂×C₄×C₆₇	1	4	yes	—
150	C₂²×Q₈	C₂×C₄×C₆₇	1	8	yes	—
151	C₂³×C₄	—	2	2	no	—
152	C₂³×C₄	C₂×C₄×C₆₇	1	1	yes	characteristic
153	C₂³×C₄	D₄×C₆₇	1	2	yes	—
154	C₂²×C₈	—	2	4	no	—
155	C₄⋊Q₈	—	2	4	no	—
156	C₄⋊Q₈	—	2	8	no	—
157	C₄⋊Q₈	—	2	8	no	—
158	C₄⋊Q₈	—	2	4	no	—
159	C₄⋊Q₈	C₂×C₄×C₆₇	1	4	yes	—
160	C₄⋊Q₈	C₂×C₄×C₆₇	1	8	yes	—
161	C₄⋊Q₈	C₂³×C₆₇	1	4	yes	—
162	C₂×C₄⋊C₄	—	2	4	no	—
163	C₂×C₄⋊C₄	—	2	8	no	—
164	C₂×C₄⋊C₄	—	2	8	no	—
165	C₂×C₄⋊C₄	—	2	4	no	—
166	C₂×C₄²	C₂×C₄×C₆₇	1	1	yes	characteristic
167	C₂×C₄²	C₂×C₄×C₆₇	1	2	yes	—
168	C₂×C₄²	D₄×C₆₇	1	2	yes	—
169	C₂×C₄²	D₄×C₆₇	1	4	yes	—
170	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
171	C₂×C₄²	C₂³×C₆₇	1	2	yes	—
172	C₄⋊C₈	—	2	4	no	—
173	C₄⋊C₈	C₂³×C₆₇	1	8	yes	—
174	C₂²⋊C₈	—	2	8	no	—
175	#97	—	2	8	no	—
176	C₂⁴	C₂²⋊C₄×C₆₇	1	1	yes	characteristic
177	C₂×Q₈	—	2	8	no	—
178	C₂×Q₈	—	2	4	no	—
179	C₂×Q₈	—	4	16	no	—
180	C₂×Q₈	—	4	8	no	—
181	C₂×Q₈	—	4	16	no	—
182	C₂×Q₈	—	4	8	no	—
183	C₂×Q₈	C₂²×C₄×C₆₇	1	8	yes	—
184	C₂²×C₄	—	2	4	no	—
185	C₂²×C₄	—	2	8	no	—
186	C₂²×C₄	—	2	8	no	—
187	C₂²×C₄	—	2	4	no	—
188	C₂²×C₄	—	4	8	no	—
189	C₂²×C₄	C₂²⋊C₄×C₆₇	1	2	yes	—
190	C₂²×C₄	C₂²⋊C₄×C₆₇	1	4	yes	—
191	C₂²×C₄	C₂²⋊C₄×C₆₇	1	2	yes	—
192	C₂²×C₄	C₂²⋊C₄×C₆₇	1	1	yes	characteristic
193	C₂²×C₄	C₂²×C₄×C₆₇	1	2	yes	—
194	C₂²×C₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
195	C₂²×C₄	C₂×D₄×C₆₇	1	4	yes	—
196	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
197	C₂×C₈	—	2	8	no	—
198	C₂×C₈	—	4	8	no	—
199	C₄⋊C₄	—	2	16	no	—
200	C₄⋊C₄	—	2	8	no	—
201	C₄⋊C₄	—	2	8	no	—
202	C₄²	C₂²⋊C₄×C₆₇	1	2	yes	—
203	C₄²	C₂²⋊C₄×C₆₇	1	4	yes	—
204	C₄²	C₂²×C₄×C₆₇	1	2	yes	—
205	C₄²	C₂²×C₄×C₆₇	1	1	yes	characteristic
206	C₄²	C₂×D₄×C₆₇	1	4	yes	—
207	C₄²	C₂×D₄×C₆₇	1	2	yes	—
208	C₄²	C₂⁴×C₆₇	1	1	yes	characteristic
209	C₂³	—	2	4	no	—
210	C₂³	#100×C₆₇	1	4	yes	—
211	C₂³	C₂×C₂²⋊C₄×C₆₇	1	2	yes	—
212	C₂³	C₂×C₂²⋊C₄×C₆₇	1	1	yes	characteristic
213	Q₈	—	4	8	no	—
214	Q₈	—	4	8	no	—
215	Q₈	—	4	16	no	—
216	C₂×C₄	—	2	4	no	—
217	C₂×C₄	—	2	2	no	—
218	C₂×C₄	—	2	4	no	—
219	C₂×C₄	—	2	2	no	—
220	C₂×C₄	—	2	4	no	—
221	C₂×C₄	—	2	8	no	—
222	C₂×C₄	—	2	8	no	—
223	C₂×C₄	—	2	8	no	—
224	C₂×C₄	—	4	16	no	—
225	C₂×C₄	—	4	8	no	—
226	C₂×C₄	C₂×C₂²⋊C₄×C₆₇	1	4	yes	—
227	C₂×C₄	C₂×C₂²⋊C₄×C₆₇	1	2	yes	—
228	C₂×C₄	C₂×C₂²⋊C₄×C₆₇	1	1	yes	characteristic
229	C₂×C₄	C₂×C₂²⋊C₄×C₆₇	1	2	yes	—
230	C₂×C₄	#137×C₆₇	1	8	yes	—
231	C₂×C₄	C₂³×C₄×C₆₇	1	1	yes	characteristic
232	C₂×C₄	C₂²×D₄×C₆₇	1	2	yes	—
233	C₈	—	4	8	no	—
234	C₂²	—	2	8	no	—
235	C₂²	—	2	4	no	—
236	C₂²	C₂×#100×C₆₇	1	4	yes	—
237	C₂²	C₂×#100×C₆₇	1	2	yes	—
238	C₂²	#419×C₆₇	1	4	yes	—
239	C₂²	C₂²×C₂²⋊C₄×C₆₇	1	1	yes	characteristic
240	C₄	—	2	1	no	—
241	C₄	—	2	2	no	—
242	C₄	—	2	1	no	—
243	C₄	—	2	4	no	—
244	C₄	—	4	8	no	—
245	C₄	#419×C₆₇	1	4	yes	—
246	C₄	C₂×#137×C₆₇	1	4	yes	—
247	C₂	—	2	4	no	—
248	C₂	C₆₇×#1558	1	4	yes	direct product factor
249	C₂	C₂²×#100×C₆₇	1	1	yes	characteristic
250	C₂	C₂×#419×C₆₇	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₆₇	C₂×C₂²⋊C₄
commutator	C₂×C₄	C₂³×C₄×C₆₇
Frattini, cosocle	C₄²	C₂⁴×C₆₇
prime radical	C₂⁴×C₆₇	C₂²⋊C₄
LCS	G ▷ C₂×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³×C₆₇ ◁ C₂²×C₄²×C₆₇ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₆₇×#1558	C₂	direct (N × H)
4. C₂×#333×C₆₇	C₂	semidirect
19. #333×C₆₇	C₂²	semidirect
121. C₂×#1558	C₆₇	direct (N × H)
123. C₂×C₆₇	#1558	direct (N × H)
128. #1558	C₂×C₆₇	direct (N × H)
129. C₂×#333	C₂×C₆₇	semidirect
130. C₆₇	C₂×#1558	direct (N × H)
145. #333	C₂²×C₆₇	semidirect
248. C₂	C₆₇×#1558	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×C₄⋊Q₈×C₆₇	2
C₂×#432×C₆₇	2
C₆₇×#1558	2
C₂×#333×C₆₇	2
C₂×#1558	67

Subgroups — C2×C67×#1558 #30696983

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₆₇×#1558 #30696983