Subgroups — C₁₀₉×#9700 #42127694

Order 27904.

478 conjugacy classes of subgroups.

The subgroup lattice has 244 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₁₀₉×#2862	C₂	1	1	yes	maximal, characteristic
2	C₁₀₉×#1831	C₂	1	1	yes	maximal, characteristic
3	C₄×C₄⋊C₈×C₁₀₉	C₂	1	1	yes	maximal, characteristic
4	C₂×#332×C₁₀₉	C₂	1	2	yes	maximal
5	C₂×#332×C₁₀₉	C₂	1	2	yes	maximal
6	C₂×C₄⋊Q₈×C₁₀₉	C₂²	1	1	yes	characteristic
7	C₂×C₄⋊Q₈×C₁₀₉	C₂²	1	1	yes	characteristic
8	C₂×C₄⋊C₈×C₁₀₉	C₂²	1	2	yes	—
9	C₂×C₄⋊C₈×C₁₀₉	C₂²	1	2	yes	—
10	C₂×C₄×C₈×C₁₀₉	—	2	1	no	—
11	#391×C₁₀₉	—	2	1	no	—
12	#391×C₁₀₉	—	2	2	no	—
13	#391×C₁₀₉	—	2	1	no	—
14	#382×C₁₀₉	C₄	1	4	yes	—
15	C₄³×C₁₀₉	C₂²	1	1	yes	characteristic
16	#336×C₁₀₉	—	2	1	no	—
17	#336×C₁₀₉	—	2	2	no	—
18	#332×C₁₀₉	—	2	4	no	—
19	#332×C₁₀₉	—	2	4	no	—
20	C₂²×Q₈×C₁₀₉	—	2	1	no	—
21	C₂²×Q₈×C₁₀₉	—	2	1	no	—
22	C₂²×C₈×C₁₀₉	—	2	2	no	—
23	C₂²×C₈×C₁₀₉	—	2	2	no	—
24	C₄⋊Q₈×C₁₀₉	—	2	2	no	—
25	C₄⋊Q₈×C₁₀₉	—	2	2	no	—
26	C₄⋊Q₈×C₁₀₉	—	2	2	no	—
27	C₄⋊Q₈×C₁₀₉	—	2	2	no	—
28	C₂×C₄⋊C₄×C₁₀₉	—	2	1	no	—
29	C₂×C₄⋊C₄×C₁₀₉	—	2	1	no	—
30	C₂×C₄⋊C₄×C₁₀₉	C₂×C₄	1	2	yes	—
31	C₂×C₄⋊C₄×C₁₀₉	C₂×C₄	1	2	yes	—
32	C₂×C₄²×C₁₀₉	C₂×C₄	1	2	yes	—
33	C₂×C₄²×C₁₀₉	D₄	1	1	yes	characteristic
34	C₂×C₄²×C₁₀₉	D₄	1	2	yes	—
35	C₂×C₄²×C₁₀₉	D₄	1	1	yes	characteristic
36	C₂×C₄²×C₁₀₉	C₂³	1	1	yes	characteristic
37	C₄⋊C₈×C₁₀₉	—	2	2	no	—
38	C₄⋊C₈×C₁₀₉	—	2	2	no	—
39	C₄⋊C₈×C₁₀₉	—	2	2	no	—
40	C₄⋊C₈×C₁₀₉	D₄	1	4	yes	—
41	C₄×C₈×C₁₀₉	—	4	2	no	—
42	#95×C₁₀₉	—	2	4	no	—
43	#95×C₁₀₉	—	2	4	no	—
44	C₂×Q₈×C₁₀₉	—	4	2	no	—
45	C₂×Q₈×C₁₀₉	—	4	1	no	—
46	C₂×Q₈×C₁₀₉	—	4	1	no	—
47	C₂×Q₈×C₁₀₉	—	4	2	no	—
48	C₂×Q₈×C₁₀₉	—	4	4	no	—
49	C₂×Q₈×C₁₀₉	—	4	1	no	—
50	C₂×Q₈×C₁₀₉	—	4	1	no	—
51	C₂²×C₄×C₁₀₉	—	2	2	no	—
52	C₂²×C₄×C₁₀₉	—	2	2	no	—
53	C₂²×C₄×C₁₀₉	C₂²⋊C₄	1	2	yes	—
54	C₂²×C₄×C₁₀₉	C₂²⋊C₄	1	2	yes	—
55	C₂²×C₄×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
56	C₂²×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
57	C₂²×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
58	C₂×C₈×C₁₀₉	—	2	4	no	—
59	C₂×C₈×C₁₀₉	—	4	2	no	—
60	C₂×C₈×C₁₀₉	—	4	2	no	—
61	C₂×C₈×C₁₀₉	—	4	2	no	—
62	C₂×C₈×C₁₀₉	—	4	4	no	—
63	C₄⋊C₄×C₁₀₉	—	2	4	no	—
64	C₄⋊C₄×C₁₀₉	—	2	4	no	—
65	C₄⋊C₄×C₁₀₉	—	4	2	no	—
66	C₄⋊C₄×C₁₀₉	—	4	1	no	—
67	C₄⋊C₄×C₁₀₉	—	4	1	no	—
68	C₄²×C₁₀₉	—	2	4	no	—
69	C₄²×C₁₀₉	—	2	4	no	—
70	C₄²×C₁₀₉	—	2	2	no	—
71	C₄²×C₁₀₉	—	2	1	no	—
72	C₄²×C₁₀₉	—	2	1	no	—
73	C₄²×C₁₀₉	C₂×D₄	1	2	yes	—
74	C₄²×C₁₀₉	#41	1	2	yes	—
75	C₂³×C₁₀₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
76	Q₈×C₁₀₉	—	8	2	no	—
77	Q₈×C₁₀₉	—	8	2	no	—
78	Q₈×C₁₀₉	—	8	2	no	—
79	Q₈×C₁₀₉	—	8	2	no	—
80	C₂×C₄×C₁₀₉	—	2	2	no	—
81	C₂×C₄×C₁₀₉	—	2	2	no	—
82	C₂×C₄×C₁₀₉	—	2	2	no	—
83	C₂×C₄×C₁₀₉	—	2	2	no	—
84	C₂×C₄×C₁₀₉	—	2	2	no	—
85	C₂×C₄×C₁₀₉	—	2	2	no	—
86	C₂×C₄×C₁₀₉	—	4	2	no	—
87	C₂×C₄×C₁₀₉	—	4	2	no	—
88	C₂×C₄×C₁₀₉	—	4	4	no	—
89	C₂×C₄×C₁₀₉	—	4	2	no	—
90	C₂×C₄×C₁₀₉	—	4	2	no	—
91	C₂×C₄×C₁₀₉	#101	1	4	yes	—
92	C₂×C₄×C₁₀₉	C₄×D₄	1	2	yes	—
93	C₂×C₄×C₁₀₉	#121	1	2	yes	—
94	C₂×C₄×C₁₀₉	#124	1	1	yes	characteristic
95	C₂×C₄×C₁₀₉	#127	1	1	yes	characteristic
96	C₂×C₄×C₁₀₉	#135	1	2	yes	—
97	C₂×C₄×C₁₀₉	#135	1	4	yes	—
98	C₂×C₄×C₁₀₉	AGL(1,ℤ/8)	1	1	yes	characteristic
99	C₂×C₄×C₁₀₉	#137	1	1	yes	characteristic
100	C₈×C₁₀₉	—	4	4	no	—
101	C₈×C₁₀₉	—	8	2	no	—
102	C₂²×C₁₀₉	#390	1	1	yes	characteristic
103	C₂²×C₁₀₉	C₂×#101	1	2	yes	—
104	C₂²×C₁₀₉	#416	1	1	yes	characteristic
105	C₂²×C₁₀₉	#416	1	2	yes	—
106	C₂²×C₁₀₉	#417	1	1	yes	characteristic
107	C₄×C₁₀₉	—	2	4	no	—
108	C₄×C₁₀₉	—	2	2	no	—
109	C₄×C₁₀₉	—	4	2	no	—
110	C₄×C₁₀₉	—	8	2	no	—
111	C₄×C₁₀₉	—	8	2	no	—
112	C₄×C₁₀₉	#463	1	2	yes	—
113	C₄×C₁₀₉	#464	1	2	yes	—
114	C₄×C₁₀₉	#488	1	2	yes	—
115	C₄×C₁₀₉	#495	1	2	yes	—
116	#9700	C₁₀₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
117	C₂×C₁₀₉	#1544	1	2	yes	—
118	C₂×C₁₀₉	#1552	1	2	yes	—
119	C₂×C₁₀₉	#1958	1	1	yes	characteristic
120	C₂×C₁₀₉	#1966	1	1	yes	characteristic
121	C₂×C₁₀₉	#1983	1	1	yes	characteristic
122	#2862	C₂×C₁₀₉	1	1	yes	characteristic
123	#1831	C₂×C₁₀₉	1	1	yes	characteristic
124	C₄×C₄⋊C₈	C₂×C₁₀₉	1	1	yes	characteristic
125	C₂×#332	C₂×C₁₀₉	1	2	yes	—
126	C₂×#332	C₂×C₁₀₉	1	2	yes	—
127	C₁₀₉	#9700	1	1	yes	characteristic, direct product factor, Sylow
128	C₂×C₄⋊Q₈	C₂²×C₁₀₉	1	1	yes	characteristic
129	C₂×C₄⋊Q₈	C₂²×C₁₀₉	1	1	yes	characteristic
130	C₂×C₄⋊C₈	C₂²×C₁₀₉	1	2	yes	—
131	C₂×C₄⋊C₈	C₂²×C₁₀₉	1	2	yes	—
132	C₂×C₄×C₈	—	2	1	no	—
133	#391	—	2	1	no	—
134	#391	—	2	2	no	—
135	#391	—	2	1	no	—
136	#382	C₄×C₁₀₉	1	4	yes	—
137	C₄³	C₂²×C₁₀₉	1	1	yes	characteristic
138	#336	—	2	2	no	—
139	#336	—	2	1	no	—
140	#332	—	2	4	no	—
141	#332	—	2	4	no	—
142	C₂²×Q₈	—	2	1	no	—
143	C₂²×Q₈	—	2	1	no	—
144	C₂²×C₈	—	2	2	no	—
145	C₂²×C₈	—	2	2	no	—
146	C₄⋊Q₈	—	2	2	no	—
147	C₄⋊Q₈	—	2	2	no	—
148	C₄⋊Q₈	—	2	2	no	—
149	C₄⋊Q₈	—	2	2	no	—
150	C₂×C₄⋊C₄	—	2	1	no	—
151	C₂×C₄⋊C₄	—	2	1	no	—
152	C₂×C₄⋊C₄	C₂×C₄×C₁₀₉	1	2	yes	—
153	C₂×C₄⋊C₄	C₂×C₄×C₁₀₉	1	2	yes	—
154	C₂×C₄²	C₂×C₄×C₁₀₉	1	2	yes	—
155	C₂×C₄²	D₄×C₁₀₉	1	1	yes	characteristic
156	C₂×C₄²	D₄×C₁₀₉	1	2	yes	—
157	C₂×C₄²	D₄×C₁₀₉	1	1	yes	characteristic
158	C₂×C₄²	C₂³×C₁₀₉	1	1	yes	characteristic
159	C₄⋊C₈	—	2	2	no	—
160	C₄⋊C₈	—	2	2	no	—
161	C₄⋊C₈	—	2	2	no	—
162	C₄⋊C₈	D₄×C₁₀₉	1	4	yes	—
163	C₄×C₈	—	4	2	no	—
164	#95	—	2	4	no	—
165	#95	—	2	4	no	—
166	C₂×Q₈	—	4	2	no	—
167	C₂×Q₈	—	4	1	no	—
168	C₂×Q₈	—	4	1	no	—
169	C₂×Q₈	—	4	2	no	—
170	C₂×Q₈	—	4	4	no	—
171	C₂×Q₈	—	4	1	no	—
172	C₂×Q₈	—	4	1	no	—
173	C₂²×C₄	—	2	2	no	—
174	C₂²×C₄	—	2	2	no	—
175	C₂²×C₄	C₂²⋊C₄×C₁₀₉	1	2	yes	—
176	C₂²×C₄	C₂²⋊C₄×C₁₀₉	1	2	yes	—
177	C₂²×C₄	C₂²×C₄×C₁₀₉	1	1	yes	characteristic
178	C₂²×C₄	C₂×D₄×C₁₀₉	1	1	yes	characteristic
179	C₂²×C₄	C₂×D₄×C₁₀₉	1	1	yes	characteristic
180	C₂×C₈	—	2	4	no	—
181	C₂×C₈	—	4	2	no	—
182	C₂×C₈	—	4	2	no	—
183	C₂×C₈	—	4	2	no	—
184	C₂×C₈	—	4	4	no	—
185	C₄⋊C₄	—	2	4	no	—
186	C₄⋊C₄	—	2	4	no	—
187	C₄⋊C₄	—	4	2	no	—
188	C₄⋊C₄	—	4	1	no	—
189	C₄⋊C₄	—	4	1	no	—
190	C₄²	—	2	4	no	—
191	C₄²	—	2	4	no	—
192	C₄²	—	2	2	no	—
193	C₄²	—	2	1	no	—
194	C₄²	—	2	1	no	—
195	C₄²	C₂×D₄×C₁₀₉	1	2	yes	—
196	C₄²	#41×C₁₀₉	1	2	yes	—
197	C₂³	C₂×C₂²⋊C₄×C₁₀₉	1	1	yes	characteristic
198	Q₈	—	8	2	no	—
199	Q₈	—	8	2	no	—
200	Q₈	—	8	2	no	—
201	Q₈	—	8	2	no	—
202	C₂×C₄	—	2	2	no	—
203	C₂×C₄	—	2	2	no	—
204	C₂×C₄	—	2	2	no	—
205	C₂×C₄	—	2	2	no	—
206	C₂×C₄	—	2	2	no	—
207	C₂×C₄	—	2	2	no	—
208	C₂×C₄	—	4	2	no	—
209	C₂×C₄	—	4	2	no	—
210	C₂×C₄	—	4	4	no	—
211	C₂×C₄	—	4	2	no	—
212	C₂×C₄	—	4	2	no	—
213	C₂×C₄	#101×C₁₀₉	1	4	yes	—
214	C₂×C₄	C₄×D₄×C₁₀₉	1	2	yes	—
215	C₂×C₄	#121×C₁₀₉	1	2	yes	—
216	C₂×C₄	#124×C₁₀₉	1	1	yes	characteristic
217	C₂×C₄	#127×C₁₀₉	1	1	yes	characteristic
218	C₂×C₄	#135×C₁₀₉	1	2	yes	—
219	C₂×C₄	#135×C₁₀₉	1	4	yes	—
220	C₂×C₄	AGL(1,ℤ/8)×C₁₀₉	1	1	yes	characteristic
221	C₂×C₄	#137×C₁₀₉	1	1	yes	characteristic
222	C₈	—	4	4	no	—
223	C₈	—	8	2	no	—
224	C₂²	#390×C₁₀₉	1	1	yes	characteristic
225	C₂²	C₂×#101×C₁₀₉	1	2	yes	—
226	C₂²	#416×C₁₀₉	1	2	yes	—
227	C₂²	#416×C₁₀₉	1	1	yes	characteristic
228	C₂²	#417×C₁₀₉	1	1	yes	characteristic
229	C₄	—	2	2	no	—
230	C₄	—	2	4	no	—
231	C₄	—	4	2	no	—
232	C₄	—	8	2	no	—
233	C₄	—	8	2	no	—
234	C₄	#463×C₁₀₉	1	2	yes	—
235	C₄	#464×C₁₀₉	1	2	yes	—
236	C₄	#488×C₁₀₉	1	2	yes	—
237	C₄	#495×C₁₀₉	1	2	yes	—
238	C₂	C₁₀₉×#1544	1	2	yes	—
239	C₂	C₁₀₉×#1552	1	2	yes	—
240	C₂	C₁₀₉×#1958	1	1	yes	characteristic
241	C₂	C₁₀₉×#1966	1	1	yes	characteristic
242	C₂	C₁₀₉×#1983	1	1	yes	characteristic

	subgroup	G/H
center, socle, prime radical	C₂³×C₁₀₉	C₂×C₂²⋊C₄
commutator	C₂²×C₄	C₂²×C₄×C₁₀₉
Frattini, cosocle	C₂×C₄²	C₂³×C₁₀₉
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³×C₁₀₉ ◁ C₄³×C₁₀₉ ◁ G

Complements

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

N	H ≅ G/N	type
116. #9700	C₁₀₉	direct (N × H)
127. C₁₀₉	#9700	direct (N × H)

Maximal subgroups

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

subgroup	index
C₁₀₉×#2862	2
C₁₀₉×#1831	2
C₄×C₄⋊C₈×C₁₀₉	2
C₂×#332×C₁₀₉	2
#9700	109

Subgroups — C109×#9700 #42127694

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₀₉×#9700 #42127694