Subgroups — C₁₀₉×#9699 #42127693

Order 27904.

602 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	C₁₀₉×#2858	C₂	1	1	yes	maximal, characteristic
2	C₁₀₉×#1831	C₂	1	1	yes	maximal, characteristic, complemented
3	C₄×C₄⋊C₈×C₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×#333×C₁₀₉	C₂	1	2	yes	maximal, complemented
5	C₂×#331×C₁₀₉	C₂	1	2	yes	maximal
6	C₂×C₄⋊Q₈×C₁₀₉	C₂²	1	1	yes	characteristic
7	C₂×#127×C₁₀₉	C₂²	1	1	yes	characteristic
8	C₂×#124×C₁₀₉	C₄	1	4	yes	—
9	C₂×C₄⋊C₈×C₁₀₉	C₂²	1	2	yes	—
10	C₂×C₄⋊C₈×C₁₀₉	C₂²	1	2	yes	—
11	C₂×C₄×C₈×C₁₀₉	—	2	1	no	—
12	#390×C₁₀₉	—	2	1	no	—
13	#390×C₁₀₉	—	2	2	no	—
14	#390×C₁₀₉	—	2	1	no	—
15	C₄³×C₁₀₉	C₂²	1	1	yes	characteristic
16	#336×C₁₀₉	—	2	1	no	—
17	#336×C₁₀₉	—	2	2	no	—
18	#333×C₁₀₉	—	2	4	no	—
19	#331×C₁₀₉	—	2	4	no	—
20	C₂²×Q₈×C₁₀₉	C₂×C₄	1	2	yes	—
21	C₂²×D₄×C₁₀₉	—	2	1	no	—
22	C₂²×D₄×C₁₀₉	—	2	1	no	—
23	C₂²×D₄×C₁₀₉	C₂×C₄	1	2	yes	—
24	C₂²×C₈×C₁₀₉	—	2	2	no	—
25	C₂²×C₈×C₁₀₉	—	2	2	no	—
26	C₄⋊Q₈×C₁₀₉	—	2	2	no	—
27	C₄⋊Q₈×C₁₀₉	—	2	2	no	—
28	#127×C₁₀₉	—	2	2	no	—
29	#127×C₁₀₉	—	2	2	no	—
30	#124×C₁₀₉	—	4	8	no	—
31	C₂×C₄⋊C₄×C₁₀₉	—	2	1	no	—
32	C₂×C₄⋊C₄×C₁₀₉	—	2	1	no	—
33	C₂×C₂²⋊C₄×C₁₀₉	—	2	4	no	—
34	C₂×C₂²⋊C₄×C₁₀₉	—	2	4	no	—
35	C₂×C₄²×C₁₀₉	C₂×C₄	1	2	yes	—
36	C₂×C₄²×C₁₀₉	D₄	1	1	yes	characteristic
37	C₂×C₄²×C₁₀₉	D₄	1	2	yes	—
38	C₂×C₄²×C₁₀₉	D₄	1	1	yes	characteristic
39	C₂×C₄²×C₁₀₉	C₂³	1	1	yes	characteristic
40	C₄⋊C₈×C₁₀₉	—	2	2	no	—
41	C₄⋊C₈×C₁₀₉	—	2	2	no	—
42	C₄⋊C₈×C₁₀₉	—	2	2	no	—
43	C₄⋊C₈×C₁₀₉	D₄	1	4	yes	—
44	C₄×C₈×C₁₀₉	—	4	2	no	—
45	C₂⁴×C₁₀₉	—	2	2	no	—
46	C₂×Q₈×C₁₀₉	—	2	4	no	—
47	C₂×Q₈×C₁₀₉	—	4	4	no	—
48	C₂×D₄×C₁₀₉	—	2	4	no	—
49	C₂×D₄×C₁₀₉	—	4	4	no	—
50	C₂×D₄×C₁₀₉	—	4	2	no	—
51	C₂×D₄×C₁₀₉	—	4	1	no	—
52	C₂×D₄×C₁₀₉	—	4	1	no	—
53	C₂×D₄×C₁₀₉	—	4	2	no	—
54	C₂×D₄×C₁₀₉	—	4	4	no	—
55	C₂×D₄×C₁₀₉	—	4	1	no	—
56	C₂×D₄×C₁₀₉	—	4	1	no	—
57	C₂²×C₄×C₁₀₉	—	2	2	no	—
58	C₂²×C₄×C₁₀₉	C₂²⋊C₄	1	2	yes	—
59	C₂²×C₄×C₁₀₉	C₂²⋊C₄	1	2	yes	—
60	C₂²×C₄×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
61	C₂²×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
62	C₂²×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
63	C₂×C₈×C₁₀₉	—	2	4	no	—
64	C₂×C₈×C₁₀₉	—	4	2	no	—
65	C₂×C₈×C₁₀₉	—	4	2	no	—
66	C₂×C₈×C₁₀₉	—	4	2	no	—
67	C₂×C₈×C₁₀₉	—	4	4	no	—
68	C₄⋊C₄×C₁₀₉	—	4	2	no	—
69	C₄⋊C₄×C₁₀₉	—	4	1	no	—
70	C₄⋊C₄×C₁₀₉	—	4	1	no	—
71	C₂²⋊C₄×C₁₀₉	—	4	8	no	—
72	C₂²⋊C₄×C₁₀₉	—	4	8	no	—
73	C₄²×C₁₀₉	—	2	4	no	—
74	C₄²×C₁₀₉	—	2	4	no	—
75	C₄²×C₁₀₉	—	2	2	no	—
76	C₄²×C₁₀₉	—	2	1	no	—
77	C₄²×C₁₀₉	—	2	1	no	—
78	C₄²×C₁₀₉	C₂×D₄	1	2	yes	—
79	C₄²×C₁₀₉	#41	1	2	yes	—
80	C₂³×C₁₀₉	—	4	2	no	—
81	C₂³×C₁₀₉	—	4	4	no	—
82	C₂³×C₁₀₉	—	4	2	no	—
83	C₂³×C₁₀₉	—	4	4	no	—
84	C₂³×C₁₀₉	—	4	2	no	—
85	C₂³×C₁₀₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
86	Q₈×C₁₀₉	—	8	4	no	—
87	D₄×C₁₀₉	—	8	4	no	—
88	D₄×C₁₀₉	—	8	2	no	—
89	D₄×C₁₀₉	—	8	2	no	—
90	D₄×C₁₀₉	—	8	2	no	—
91	D₄×C₁₀₉	—	8	2	no	—
92	C₂×C₄×C₁₀₉	—	2	2	no	—
93	C₂×C₄×C₁₀₉	—	2	2	no	—
94	C₂×C₄×C₁₀₉	—	2	2	no	—
95	C₂×C₄×C₁₀₉	—	2	2	no	—
96	C₂×C₄×C₁₀₉	—	2	2	no	—
97	C₂×C₄×C₁₀₉	—	2	2	no	—
98	C₂×C₄×C₁₀₉	—	4	2	no	—
99	C₂×C₄×C₁₀₉	—	4	4	no	—
100	C₂×C₄×C₁₀₉	#100	1	4	yes	—
101	C₂×C₄×C₁₀₉	C₄×D₄	1	2	yes	—
102	C₂×C₄×C₁₀₉	#121	1	2	yes	—
103	C₂×C₄×C₁₀₉	#124	1	1	yes	characteristic
104	C₂×C₄×C₁₀₉	#127	1	1	yes	characteristic
105	C₂×C₄×C₁₀₉	#135	1	2	yes	—
106	C₂×C₄×C₁₀₉	#135	1	4	yes	—
107	C₂×C₄×C₁₀₉	AGL(1,ℤ/8)	1	1	yes	characteristic
108	C₂×C₄×C₁₀₉	#137	1	1	yes	characteristic
109	C₈×C₁₀₉	—	4	4	no	—
110	C₈×C₁₀₉	—	8	2	no	—
111	C₂²×C₁₀₉	—	8	4	no	—
112	C₂²×C₁₀₉	—	8	4	no	—
113	C₂²×C₁₀₉	—	8	2	no	—
114	C₂²×C₁₀₉	—	8	2	no	—
115	C₂²×C₁₀₉	—	8	2	no	—
116	C₂²×C₁₀₉	#390	1	1	yes	characteristic
117	C₂²×C₁₀₉	C₂×#100	1	2	yes	—
118	C₂²×C₁₀₉	#416	1	1	yes	characteristic
119	C₂²×C₁₀₉	#416	1	2	yes	—
120	C₂²×C₁₀₉	#417	1	1	yes	characteristic
121	C₄×C₁₀₉	—	2	4	no	—
122	C₄×C₁₀₉	—	2	2	no	—
123	C₄×C₁₀₉	—	4	2	no	—
124	C₄×C₁₀₉	—	8	2	no	—
125	C₄×C₁₀₉	#463	1	2	yes	—
126	C₄×C₁₀₉	#464	1	2	yes	—
127	C₄×C₁₀₉	#488	1	2	yes	—
128	C₄×C₁₀₉	#495	1	2	yes	—
129	#9699	C₁₀₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
130	C₂×C₁₀₉	—	16	2	no	—
131	C₂×C₁₀₉	#1543	1	2	yes	—
132	C₂×C₁₀₉	#1551	1	2	yes	—
133	C₂×C₁₀₉	#1958	1	1	yes	characteristic
134	C₂×C₁₀₉	#1965	1	1	yes	characteristic
135	C₂×C₁₀₉	#1983	1	1	yes	characteristic
136	#2858	C₂×C₁₀₉	1	1	yes	characteristic
137	#1831	C₂×C₁₀₉	1	1	yes	characteristic, complemented
138	C₄×C₄⋊C₈	C₂×C₁₀₉	1	1	yes	characteristic, complemented
139	C₂×#333	C₂×C₁₀₉	1	2	yes	complemented
140	C₂×#331	C₂×C₁₀₉	1	2	yes	—
141	C₁₀₉	#9699	1	1	yes	characteristic, direct product factor, Sylow
142	C₂×C₄⋊Q₈	C₂²×C₁₀₉	1	1	yes	characteristic
143	C₂×#127	C₂²×C₁₀₉	1	1	yes	characteristic
144	C₂×#124	C₄×C₁₀₉	1	4	yes	—
145	C₂×C₄⋊C₈	C₂²×C₁₀₉	1	2	yes	—
146	C₂×C₄⋊C₈	C₂²×C₁₀₉	1	2	yes	—
147	C₂×C₄×C₈	—	2	1	no	—
148	#390	—	2	1	no	—
149	#390	—	2	2	no	—
150	#390	—	2	1	no	—
151	C₄³	C₂²×C₁₀₉	1	1	yes	characteristic
152	#336	—	2	2	no	—
153	#336	—	2	1	no	—
154	#333	—	2	4	no	—
155	#331	—	2	4	no	—
156	C₂²×Q₈	C₂×C₄×C₁₀₉	1	2	yes	—
157	C₂²×D₄	—	2	1	no	—
158	C₂²×D₄	—	2	1	no	—
159	C₂²×D₄	C₂×C₄×C₁₀₉	1	2	yes	—
160	C₂²×C₈	—	2	2	no	—
161	C₂²×C₈	—	2	2	no	—
162	C₄⋊Q₈	—	2	2	no	—
163	C₄⋊Q₈	—	2	2	no	—
164	#127	—	2	2	no	—
165	#127	—	2	2	no	—
166	#124	—	4	8	no	—
167	C₂×C₄⋊C₄	—	2	1	no	—
168	C₂×C₄⋊C₄	—	2	1	no	—
169	C₂×C₂²⋊C₄	—	2	4	no	—
170	C₂×C₂²⋊C₄	—	2	4	no	—
171	C₂×C₄²	C₂×C₄×C₁₀₉	1	2	yes	—
172	C₂×C₄²	D₄×C₁₀₉	1	1	yes	characteristic
173	C₂×C₄²	D₄×C₁₀₉	1	2	yes	—
174	C₂×C₄²	D₄×C₁₀₉	1	1	yes	characteristic
175	C₂×C₄²	C₂³×C₁₀₉	1	1	yes	characteristic
176	C₄⋊C₈	—	2	2	no	—
177	C₄⋊C₈	—	2	2	no	—
178	C₄⋊C₈	—	2	2	no	—
179	C₄⋊C₈	D₄×C₁₀₉	1	4	yes	—
180	C₄×C₈	—	4	2	no	—
181	C₂⁴	—	2	2	no	—
182	C₂×Q₈	—	2	4	no	—
183	C₂×Q₈	—	4	4	no	—
184	C₂×D₄	—	2	4	no	—
185	C₂×D₄	—	4	4	no	—
186	C₂×D₄	—	4	2	no	—
187	C₂×D₄	—	4	1	no	—
188	C₂×D₄	—	4	1	no	—
189	C₂×D₄	—	4	2	no	—
190	C₂×D₄	—	4	4	no	—
191	C₂×D₄	—	4	1	no	—
192	C₂×D₄	—	4	1	no	—
193	C₂²×C₄	—	2	2	no	—
194	C₂²×C₄	C₂²⋊C₄×C₁₀₉	1	2	yes	—
195	C₂²×C₄	C₂²⋊C₄×C₁₀₉	1	2	yes	—
196	C₂²×C₄	C₂²×C₄×C₁₀₉	1	1	yes	characteristic
197	C₂²×C₄	C₂×D₄×C₁₀₉	1	1	yes	characteristic
198	C₂²×C₄	C₂×D₄×C₁₀₉	1	1	yes	characteristic
199	C₂×C₈	—	2	4	no	—
200	C₂×C₈	—	4	2	no	—
201	C₂×C₈	—	4	2	no	—
202	C₂×C₈	—	4	2	no	—
203	C₂×C₈	—	4	4	no	—
204	C₄⋊C₄	—	4	2	no	—
205	C₄⋊C₄	—	4	1	no	—
206	C₄⋊C₄	—	4	1	no	—
207	C₂²⋊C₄	—	4	8	no	—
208	C₂²⋊C₄	—	4	8	no	—
209	C₄²	—	2	4	no	—
210	C₄²	—	2	4	no	—
211	C₄²	—	2	2	no	—
212	C₄²	—	2	1	no	—
213	C₄²	—	2	1	no	—
214	C₄²	C₂×D₄×C₁₀₉	1	2	yes	—
215	C₄²	#41×C₁₀₉	1	2	yes	—
216	C₂³	—	4	2	no	—
217	C₂³	—	4	4	no	—
218	C₂³	—	4	4	no	—
219	C₂³	—	4	2	no	—
220	C₂³	—	4	2	no	—
221	C₂³	C₂×C₂²⋊C₄×C₁₀₉	1	1	yes	characteristic
222	Q₈	—	8	4	no	—
223	D₄	—	8	4	no	—
224	D₄	—	8	2	no	—
225	D₄	—	8	2	no	—
226	D₄	—	8	2	no	—
227	D₄	—	8	2	no	—
228	C₂×C₄	—	2	2	no	—
229	C₂×C₄	—	2	2	no	—
230	C₂×C₄	—	2	2	no	—
231	C₂×C₄	—	2	2	no	—
232	C₂×C₄	—	2	2	no	—
233	C₂×C₄	—	2	2	no	—
234	C₂×C₄	—	4	2	no	—
235	C₂×C₄	—	4	4	no	—
236	C₂×C₄	#100×C₁₀₉	1	4	yes	—
237	C₂×C₄	C₄×D₄×C₁₀₉	1	2	yes	—
238	C₂×C₄	#121×C₁₀₉	1	2	yes	—
239	C₂×C₄	#124×C₁₀₉	1	1	yes	characteristic
240	C₂×C₄	#127×C₁₀₉	1	1	yes	characteristic
241	C₂×C₄	#135×C₁₀₉	1	2	yes	—
242	C₂×C₄	#135×C₁₀₉	1	4	yes	—
243	C₂×C₄	AGL(1,ℤ/8)×C₁₀₉	1	1	yes	characteristic
244	C₂×C₄	#137×C₁₀₉	1	1	yes	characteristic
245	C₈	—	4	4	no	—
246	C₈	—	8	2	no	—
247	C₂²	—	8	2	no	—
248	C₂²	—	8	2	no	—
249	C₂²	—	8	2	no	—
250	C₂²	—	8	4	no	—
251	C₂²	—	8	4	no	—
252	C₂²	#390×C₁₀₉	1	1	yes	characteristic
253	C₂²	C₂×#100×C₁₀₉	1	2	yes	—
254	C₂²	#416×C₁₀₉	1	2	yes	—
255	C₂²	#416×C₁₀₉	1	1	yes	characteristic
256	C₂²	#417×C₁₀₉	1	1	yes	characteristic
257	C₄	—	2	2	no	—
258	C₄	—	2	4	no	—
259	C₄	—	4	2	no	—
260	C₄	—	8	2	no	—
261	C₄	#463×C₁₀₉	1	2	yes	—
262	C₄	#464×C₁₀₉	1	2	yes	—
263	C₄	#488×C₁₀₉	1	2	yes	—
264	C₄	#495×C₁₀₉	1	2	yes	—
265	C₂	—	16	2	no	—
266	C₂	C₁₀₉×#1543	1	2	yes	—
267	C₂	C₁₀₉×#1551	1	2	yes	—
268	C₂	C₁₀₉×#1958	1	1	yes	characteristic
269	C₂	C₁₀₉×#1965	1	1	yes	characteristic
270	C₂	C₁₀₉×#1983	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₁₀₉	C₂×C₂²⋊C₄
commutator	C₂²×C₄	C₂²×C₄×C₁₀₉
Frattini, cosocle	C₂×C₄²	C₂³×C₁₀₉
prime radical	C₂×#127×C₁₀₉	C₂²
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
2. C₁₀₉×#1831	C₂	semidirect
3. C₄×C₄⋊C₈×C₁₀₉	C₂	semidirect
4. C₂×#333×C₁₀₉	C₂	semidirect
129. #9699	C₁₀₉	direct (N × H)
137. #1831	C₂×C₁₀₉	semidirect
138. C₄×C₄⋊C₈	C₂×C₁₀₉	semidirect
139. C₂×#333	C₂×C₁₀₉	semidirect
141. C₁₀₉	#9699	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×#2858	2
C₁₀₉×#1831	2
C₄×C₄⋊C₈×C₁₀₉	2
C₂×#333×C₁₀₉	2
C₂×#331×C₁₀₉	2
#9699	109

Subgroups — C109×#9699 #42127693

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₀₉×#9699 #42127693