Group #42127393 — subgroups

Subgroups — C₁₀₉×#9399 #42127393

Order 27904.

514 conjugacy classes of subgroups; only the 138 normal subgroups are shown.

normal subgroups only

This group has 514 conjugacy classes of subgroups — too many to draw. Only its 138 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₁₀₉×#2975	C₂	1	1	yes	maximal, characteristic
2	C₁₀₉×#2563	C₂	1	1	yes	maximal, characteristic
3	C₁₀₉×#1800	C₂	1	1	yes	maximal, characteristic
4	C₁₀₉×#1548	C₂	1	2	yes	maximal, complemented
5	C₁₀₉×#1303	C₂	1	2	yes	maximal, complemented
6	C₂×C₄⋊Q₈×C₁₀₉	C₂²	1	1	yes	characteristic
7	C₂×C₄×D₄×C₁₀₉	C₂²	1	1	yes	characteristic
8	#436×C₁₀₉	—	2	4	no	—
9	#431×C₁₀₉	C₂²	1	2	yes	—
10	#423×C₁₀₉	C₂²	1	2	yes	—
11	C₂×#100×C₁₀₉	—	2	1	no	—
12	#407×C₁₀₉	—	2	2	no	—
13	#393×C₁₀₉	C₄	1	4	yes	—
14	#390×C₁₀₉	—	2	1	no	—
15	#382×C₁₀₉	—	2	1	no	—
16	#380×C₁₀₉	—	2	2	no	—
17	C₄×C₄⋊C₄×C₁₀₉	C₂²	1	1	yes	characteristic
18	#343×C₁₀₉	—	2	1	no	—
19	#333×C₁₀₉	—	2	4	no	—
20	C₂²×Q₈×C₁₀₉	C₂×C₄	1	2	yes	—
21	C₂²×D₄×C₁₀₉	D₄	1	1	yes	characteristic
22	C₂³×C₄×C₁₀₉	D₄	1	2	yes	—
23	C₂×M₁₆×C₁₀₉	—	2	4	no	—
24	C₂×M₁₆×C₁₀₉	—	2	2	no	—
25	C₄⋊Q₈×C₁₀₉	—	2	2	no	—
26	C₄⋊Q₈×C₁₀₉	—	2	2	no	—
27	C₄×D₄×C₁₀₉	—	2	2	no	—
28	C₄×D₄×C₁₀₉	—	2	2	no	—
29	C₂×C₄⋊C₄×C₁₀₉	—	2	1	no	—
30	C₂×C₄⋊C₄×C₁₀₉	—	2	1	no	—
31	C₂×C₄⋊C₄×C₁₀₉	—	2	1	no	—
32	C₂×C₄⋊C₄×C₁₀₉	D₄	1	1	yes	characteristic
33	C₂×C₂²⋊C₄×C₁₀₉	—	2	4	no	—
34	C₂×C₂²⋊C₄×C₁₀₉	C₂×C₄	1	2	yes	—
35	C₂×C₄²×C₁₀₉	—	2	1	no	—
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	D₄	1	4	yes	—
40	#100×C₁₀₉	—	4	2	no	—
41	C₂²⋊C₈×C₁₀₉	—	2	4	no	—
42	C₂²⋊C₈×C₁₀₉	—	2	2	no	—
43	C₄×C₈×C₁₀₉	—	2	2	no	—
44	#95×C₁₀₉	—	2	4	no	—
45	#95×C₁₀₉	C₂×C₄	1	2	yes	—
46	C₂⁴×C₁₀₉	C₂²⋊C₄	1	2	yes	—
47	C₂×Q₈×C₁₀₉	—	2	4	no	—
48	C₂×Q₈×C₁₀₉	—	8	2	no	—
49	C₂×D₄×C₁₀₉	—	2	2	no	—
50	C₂×D₄×C₁₀₉	—	2	1	no	—
51	C₂×D₄×C₁₀₉	—	2	2	no	—
52	C₂×D₄×C₁₀₉	—	2	1	no	—
53	C₂²×C₄×C₁₀₉	—	2	2	no	—
54	C₂²×C₄×C₁₀₉	—	2	2	no	—
55	C₂²×C₄×C₁₀₉	—	2	2	no	—
56	C₂²×C₄×C₁₀₉	—	2	2	no	—
57	C₂²×C₄×C₁₀₉	—	4	4	no	—
58	C₂²×C₄×C₁₀₉	C₂²⋊C₄	1	2	yes	—
59	C₂²×C₄×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
60	C₂²×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
61	C₂²×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
62	M₁₆×C₁₀₉	—	4	2	no	—
63	M₁₆×C₁₀₉	—	4	2	no	—
64	M₁₆×C₁₀₉	—	8	4	no	—
65	C₂×C₈×C₁₀₉	—	2	4	no	—
66	C₂×C₈×C₁₀₉	—	4	2	no	—
67	C₄⋊C₄×C₁₀₉	—	2	2	no	—
68	C₄⋊C₄×C₁₀₉	—	4	2	no	—
69	C₄⋊C₄×C₁₀₉	—	4	1	no	—
70	C₄⋊C₄×C₁₀₉	—	4	2	no	—
71	C₄⋊C₄×C₁₀₉	—	4	1	no	—
72	C₂²⋊C₄×C₁₀₉	—	2	4	no	—
73	C₂²⋊C₄×C₁₀₉	—	8	4	no	—
74	C₄²×C₁₀₉	—	4	1	no	—
75	C₄²×C₁₀₉	—	4	1	no	—
76	C₄²×C₁₀₉	C₂×D₄	1	2	yes	—
77	C₄²×C₁₀₉	#41	1	2	yes	—
78	C₂³×C₁₀₉	—	2	2	no	—
79	C₂³×C₁₀₉	—	2	2	no	—
80	C₂³×C₁₀₉	—	2	2	no	—
81	C₂³×C₁₀₉	—	4	2	no	—
82	C₂³×C₁₀₉	—	4	2	no	—
83	C₂³×C₁₀₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
84	Q₈×C₁₀₉	—	8	4	no	—
85	D₄×C₁₀₉	—	4	2	no	—
86	D₄×C₁₀₉	—	4	2	no	—
87	C₂×C₄×C₁₀₉	—	2	2	no	—
88	C₂×C₄×C₁₀₉	—	2	2	no	—
89	C₂×C₄×C₁₀₉	—	4	1	no	—
90	C₂×C₄×C₁₀₉	—	4	4	no	—
91	C₂×C₄×C₁₀₉	—	4	2	no	—
92	C₂×C₄×C₁₀₉	—	4	2	no	—
93	C₂×C₄×C₁₀₉	—	4	4	no	—
94	C₂×C₄×C₁₀₉	—	4	2	no	—
95	C₂×C₄×C₁₀₉	—	4	2	no	—
96	C₂×C₄×C₁₀₉	—	4	2	no	—
97	C₂×C₄×C₁₀₉	—	8	2	no	—
98	C₂×C₄×C₁₀₉	C₄×D₄	1	2	yes	—
99	C₂×C₄×C₁₀₉	#120	1	1	yes	characteristic
100	C₂×C₄×C₁₀₉	#121	1	2	yes	—
101	C₂×C₄×C₁₀₉	#123	1	1	yes	characteristic
102	C₂×C₄×C₁₀₉	AGL(1,ℤ/8)	1	1	yes	characteristic
103	C₂×C₄×C₁₀₉	#137	1	2	yes	—
104	C₂×C₄×C₁₀₉	#137	1	1	yes	characteristic
105	C₈×C₁₀₉	—	4	4	no	—
106	C₈×C₁₀₉	—	8	2	no	—
107	C₂²×C₁₀₉	—	2	1	no	—
108	C₂²×C₁₀₉	—	2	1	no	—
109	C₂²×C₁₀₉	—	4	4	no	—
110	C₂²×C₁₀₉	—	4	2	no	—
111	C₂²×C₁₀₉	—	4	2	no	—
112	C₂²×C₁₀₉	—	4	2	no	—
113	C₂²×C₁₀₉	—	8	2	no	—
114	C₂²×C₁₀₉	#386	1	1	yes	characteristic
115	C₂²×C₁₀₉	#417	1	1	yes	characteristic
116	C₂²×C₁₀₉	#419	1	1	yes	characteristic
117	C₄×C₁₀₉	—	2	1	no	—
118	C₄×C₁₀₉	—	2	2	no	—
119	C₄×C₁₀₉	—	4	2	no	—
120	C₄×C₁₀₉	—	8	2	no	—
121	C₄×C₁₀₉	—	8	2	no	—
122	C₄×C₁₀₉	#455	1	2	yes	—
123	C₄×C₁₀₉	#468	1	2	yes	—
124	C₄×C₁₀₉	#470	1	2	yes	—
125	#9399	C₁₀₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
126	C₂×C₁₀₉	—	2	1	no	—
127	C₂×C₁₀₉	—	2	1	no	—
128	C₂×C₁₀₉	—	8	2	no	—
129	C₂×C₁₀₉	#1885	1	1	yes	characteristic
130	C₂×C₁₀₉	#1894	1	1	yes	characteristic
131	C₂×C₁₀₉	#1914	1	1	yes	characteristic
132	#2975	C₂×C₁₀₉	1	1	yes	characteristic
133	#2563	C₂×C₁₀₉	1	1	yes	characteristic
134	#1800	C₂×C₁₀₉	1	1	yes	characteristic
135	#1548	C₂×C₁₀₉	1	2	yes	complemented
136	#1303	C₂×C₁₀₉	1	2	yes	complemented
137	C₁₀₉	#9399	1	1	yes	characteristic, direct product factor, Sylow
138	C₂×C₄⋊Q₈	C₂²×C₁₀₉	1	1	yes	characteristic
139	C₂×C₄×D₄	C₂²×C₁₀₉	1	1	yes	characteristic
140	#436	—	2	4	no	—
141	#431	C₂²×C₁₀₉	1	2	yes	—
142	#423	C₂²×C₁₀₉	1	2	yes	—
143	C₂×#100	—	2	1	no	—
144	#407	—	2	2	no	—
145	#393	C₄×C₁₀₉	1	4	yes	—
146	#390	—	2	1	no	—
147	#382	—	2	1	no	—
148	#380	—	2	2	no	—
149	C₄×C₄⋊C₄	C₂²×C₁₀₉	1	1	yes	characteristic
150	#343	—	2	1	no	—
151	#333	—	2	4	no	—
152	C₂²×Q₈	C₂×C₄×C₁₀₉	1	2	yes	—
153	C₂²×D₄	D₄×C₁₀₉	1	1	yes	characteristic
154	C₂³×C₄	D₄×C₁₀₉	1	2	yes	—
155	C₂×M₁₆	—	2	4	no	—
156	C₂×M₁₆	—	2	2	no	—
157	C₄⋊Q₈	—	2	2	no	—
158	C₄⋊Q₈	—	2	2	no	—
159	C₄×D₄	—	2	2	no	—
160	C₄×D₄	—	2	2	no	—
161	C₂×C₄⋊C₄	—	2	1	no	—
162	C₂×C₄⋊C₄	—	2	1	no	—
163	C₂×C₄⋊C₄	—	2	1	no	—
164	C₂×C₄⋊C₄	D₄×C₁₀₉	1	1	yes	characteristic
165	C₂×C₂²⋊C₄	—	2	4	no	—
166	C₂×C₂²⋊C₄	C₂×C₄×C₁₀₉	1	2	yes	—
167	C₂×C₄²	—	2	1	no	—
168	C₂×C₄²	C₂³×C₁₀₉	1	1	yes	characteristic
169	C₄⋊C₈	—	2	2	no	—
170	C₄⋊C₈	—	2	2	no	—
171	C₄⋊C₈	D₄×C₁₀₉	1	4	yes	—
172	#100	—	4	2	no	—
173	C₂²⋊C₈	—	2	4	no	—
174	C₂²⋊C₈	—	2	2	no	—
175	C₄×C₈	—	2	2	no	—
176	#95	—	2	4	no	—
177	#95	C₂×C₄×C₁₀₉	1	2	yes	—
178	C₂⁴	C₂²⋊C₄×C₁₀₉	1	2	yes	—
179	C₂×Q₈	—	2	4	no	—
180	C₂×Q₈	—	8	2	no	—
181	C₂×D₄	—	2	2	no	—
182	C₂×D₄	—	2	1	no	—
183	C₂×D₄	—	2	2	no	—
184	C₂×D₄	—	2	1	no	—
185	C₂²×C₄	—	2	2	no	—
186	C₂²×C₄	—	2	2	no	—
187	C₂²×C₄	—	2	2	no	—
188	C₂²×C₄	—	2	2	no	—
189	C₂²×C₄	—	4	4	no	—
190	C₂²×C₄	C₂²⋊C₄×C₁₀₉	1	2	yes	—
191	C₂²×C₄	C₂²×C₄×C₁₀₉	1	1	yes	characteristic
192	C₂²×C₄	C₂×D₄×C₁₀₉	1	1	yes	characteristic
193	C₂²×C₄	C₂×D₄×C₁₀₉	1	1	yes	characteristic
194	M₁₆	—	4	2	no	—
195	M₁₆	—	4	2	no	—
196	M₁₆	—	8	4	no	—
197	C₂×C₈	—	2	4	no	—
198	C₂×C₈	—	4	2	no	—
199	C₄⋊C₄	—	2	2	no	—
200	C₄⋊C₄	—	4	2	no	—
201	C₄⋊C₄	—	4	1	no	—
202	C₄⋊C₄	—	4	2	no	—
203	C₄⋊C₄	—	4	1	no	—
204	C₂²⋊C₄	—	2	4	no	—
205	C₂²⋊C₄	—	8	4	no	—
206	C₄²	—	4	1	no	—
207	C₄²	—	4	1	no	—
208	C₄²	C₂×D₄×C₁₀₉	1	2	yes	—
209	C₄²	#41×C₁₀₉	1	2	yes	—
210	C₂³	—	2	2	no	—
211	C₂³	—	2	2	no	—
212	C₂³	—	2	2	no	—
213	C₂³	—	4	2	no	—
214	C₂³	—	4	2	no	—
215	C₂³	C₂×C₂²⋊C₄×C₁₀₉	1	1	yes	characteristic
216	Q₈	—	8	4	no	—
217	D₄	—	4	2	no	—
218	D₄	—	4	2	no	—
219	C₂×C₄	—	2	2	no	—
220	C₂×C₄	—	2	2	no	—
221	C₂×C₄	—	4	1	no	—
222	C₂×C₄	—	4	4	no	—
223	C₂×C₄	—	4	2	no	—
224	C₂×C₄	—	4	2	no	—
225	C₂×C₄	—	4	4	no	—
226	C₂×C₄	—	4	2	no	—
227	C₂×C₄	—	4	2	no	—
228	C₂×C₄	—	4	2	no	—
229	C₂×C₄	—	8	2	no	—
230	C₂×C₄	C₄×D₄×C₁₀₉	1	2	yes	—
231	C₂×C₄	#120×C₁₀₉	1	1	yes	characteristic
232	C₂×C₄	#121×C₁₀₉	1	2	yes	—
233	C₂×C₄	#123×C₁₀₉	1	1	yes	characteristic
234	C₂×C₄	AGL(1,ℤ/8)×C₁₀₉	1	1	yes	characteristic
235	C₂×C₄	#137×C₁₀₉	1	2	yes	—
236	C₂×C₄	#137×C₁₀₉	1	1	yes	characteristic
237	C₈	—	4	4	no	—
238	C₈	—	8	2	no	—
239	C₂²	—	2	1	no	—
240	C₂²	—	2	1	no	—
241	C₂²	—	4	4	no	—
242	C₂²	—	4	2	no	—
243	C₂²	—	4	2	no	—
244	C₂²	—	4	2	no	—
245	C₂²	—	8	2	no	—
246	C₂²	#386×C₁₀₉	1	1	yes	characteristic
247	C₂²	#417×C₁₀₉	1	1	yes	characteristic
248	C₂²	#419×C₁₀₉	1	1	yes	characteristic
249	C₄	—	2	2	no	—
250	C₄	—	2	1	no	—
251	C₄	—	4	2	no	—
252	C₄	—	8	2	no	—
253	C₄	—	8	2	no	—
254	C₄	#455×C₁₀₉	1	2	yes	—
255	C₄	#468×C₁₀₉	1	2	yes	—
256	C₄	#470×C₁₀₉	1	2	yes	—
257	C₂	—	2	1	no	—
258	C₂	—	2	1	no	—
259	C₂	—	8	2	no	—
260	C₂	C₁₀₉×#1885	1	1	yes	characteristic
261	C₂	C₁₀₉×#1894	1	1	yes	characteristic
262	C₂	C₁₀₉×#1914	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₁₀₉×#2975

C₂

yes

maximal, characteristic

C₁₀₉×#2563

C₂

yes

maximal, characteristic

C₁₀₉×#1800

C₂

yes

maximal, characteristic

C₁₀₉×#1548

C₂

yes

maximal, complemented

C₁₀₉×#1303

C₂

yes

maximal, complemented

C₂×C₄⋊Q₈×C₁₀₉

C₂²

yes

characteristic

C₂×C₄×D₄×C₁₀₉

C₂²

yes

characteristic

#436×C₁₀₉

—

#431×C₁₀₉

C₂²

yes

—

#423×C₁₀₉

C₂²

yes

—

C₂×#100×C₁₀₉

—

#407×C₁₀₉

—

#393×C₁₀₉

C₄

yes

—

#390×C₁₀₉

—

#382×C₁₀₉

—

#380×C₁₀₉

—

C₄×C₄⋊C₄×C₁₀₉

C₂²

yes

characteristic

#343×C₁₀₉

—

#333×C₁₀₉

—

C₂²×Q₈×C₁₀₉

C₂×C₄

yes

—

C₂²×D₄×C₁₀₉

D₄

yes

characteristic

C₂³×C₄×C₁₀₉

D₄

yes

—

C₂×M₁₆×C₁₀₉

—

C₂×M₁₆×C₁₀₉

—

C₄⋊Q₈×C₁₀₉

—

C₄⋊Q₈×C₁₀₉

—

C₄×D₄×C₁₀₉

—

C₄×D₄×C₁₀₉

—

C₂×C₄⋊C₄×C₁₀₉

—

C₂×C₄⋊C₄×C₁₀₉

—

C₂×C₄⋊C₄×C₁₀₉

—

C₂×C₄⋊C₄×C₁₀₉

D₄

yes

characteristic

C₂×C₂²⋊C₄×C₁₀₉

—

C₂×C₂²⋊C₄×C₁₀₉

C₂×C₄

yes

—

C₂×C₄²×C₁₀₉

—

C₂×C₄²×C₁₀₉

C₂³

yes

characteristic

C₄⋊C₈×C₁₀₉

—

C₄⋊C₈×C₁₀₉

—

C₄⋊C₈×C₁₀₉

D₄

yes

—

#100×C₁₀₉

—

C₂²⋊C₈×C₁₀₉

—

C₂²⋊C₈×C₁₀₉

—

C₄×C₈×C₁₀₉

—

#95×C₁₀₉

—

#95×C₁₀₉

C₂×C₄

yes

—

C₂⁴×C₁₀₉

C₂²⋊C₄

yes

—

C₂×Q₈×C₁₀₉

—

C₂×Q₈×C₁₀₉

—

C₂×D₄×C₁₀₉

—

C₂×D₄×C₁₀₉

—

C₂×D₄×C₁₀₉

—

C₂×D₄×C₁₀₉

—

C₂²×C₄×C₁₀₉

—

C₂²×C₄×C₁₀₉

—

C₂²×C₄×C₁₀₉

—

C₂²×C₄×C₁₀₉

—

C₂²×C₄×C₁₀₉

—

C₂²×C₄×C₁₀₉

C₂²⋊C₄

yes

—

C₂²×C₄×C₁₀₉

C₂²×C₄

yes

characteristic

C₂²×C₄×C₁₀₉

C₂×D₄

yes

characteristic

C₂²×C₄×C₁₀₉

C₂×D₄

yes

characteristic

M₁₆×C₁₀₉

—

M₁₆×C₁₀₉

—

M₁₆×C₁₀₉

—

C₂×C₈×C₁₀₉

—

C₂×C₈×C₁₀₉

—

C₄⋊C₄×C₁₀₉

—

C₄⋊C₄×C₁₀₉

—

C₄⋊C₄×C₁₀₉

—

C₄⋊C₄×C₁₀₉

—

C₄⋊C₄×C₁₀₉

—

C₂²⋊C₄×C₁₀₉

—

C₂²⋊C₄×C₁₀₉

—

C₄²×C₁₀₉

—

C₄²×C₁₀₉

—

C₄²×C₁₀₉

C₂×D₄

yes

—

C₄²×C₁₀₉

#41

yes

—

C₂³×C₁₀₉

—

C₂³×C₁₀₉

—

C₂³×C₁₀₉

—

C₂³×C₁₀₉

—

C₂³×C₁₀₉

—

C₂³×C₁₀₉

C₂×C₂²⋊C₄

yes

characteristic

Q₈×C₁₀₉

—

D₄×C₁₀₉

—

D₄×C₁₀₉

—

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

C₄×D₄

yes

—

C₂×C₄×C₁₀₉

#120

yes

characteristic

100

C₂×C₄×C₁₀₉

#121

yes

—

101

C₂×C₄×C₁₀₉

#123

yes

characteristic

102

C₂×C₄×C₁₀₉

AGL(1,ℤ/8)

yes

characteristic

103

C₂×C₄×C₁₀₉

#137

yes

—

104

C₂×C₄×C₁₀₉

#137

yes

characteristic

105

C₈×C₁₀₉

—

106

C₈×C₁₀₉

—

107

C₂²×C₁₀₉

—

108

C₂²×C₁₀₉

—

109

C₂²×C₁₀₉

—

110

C₂²×C₁₀₉

—

111

C₂²×C₁₀₉

—

112

C₂²×C₁₀₉

—

113

C₂²×C₁₀₉

—

114

C₂²×C₁₀₉

#386

yes

characteristic

115

C₂²×C₁₀₉

#417

yes

characteristic

116

C₂²×C₁₀₉

#419

yes

characteristic

117

C₄×C₁₀₉

—

118

C₄×C₁₀₉

—

119

C₄×C₁₀₉

—

120

C₄×C₁₀₉

—

121

C₄×C₁₀₉

—

122

C₄×C₁₀₉

#455

yes

—

123

C₄×C₁₀₉

#468

yes

—

124

C₄×C₁₀₉

#470

yes

—

125

#9399

C₁₀₉

yes

maximal, characteristic, direct product factor, Sylow

126

C₂×C₁₀₉

—

127

C₂×C₁₀₉

—

128

C₂×C₁₀₉

—

129

C₂×C₁₀₉

#1885

yes

characteristic

130

C₂×C₁₀₉

#1894

yes

characteristic

131

C₂×C₁₀₉

#1914

yes

characteristic

132

#2975

C₂×C₁₀₉

yes

characteristic

133

#2563

C₂×C₁₀₉

yes

characteristic

134

#1800

C₂×C₁₀₉

yes

characteristic

135

#1548

C₂×C₁₀₉

yes

complemented

136

#1303

C₂×C₁₀₉

yes

complemented

137

C₁₀₉

#9399

yes

characteristic, direct product factor, Sylow

138

C₂×C₄⋊Q₈

C₂²×C₁₀₉

yes

characteristic

139

C₂×C₄×D₄

C₂²×C₁₀₉

yes

characteristic

140

#436

—

141

#431

C₂²×C₁₀₉

yes

—

142

#423

C₂²×C₁₀₉

yes

—

143

C₂×#100

—

144

#407

—

145

#393

C₄×C₁₀₉

yes

—

146

#390

—

147

#382

—

148

#380

—

149

C₄×C₄⋊C₄

C₂²×C₁₀₉

yes

characteristic

150

#343

—

151

#333

—

152

C₂²×Q₈

C₂×C₄×C₁₀₉

yes

—

153

C₂²×D₄

D₄×C₁₀₉

yes

characteristic

154

C₂³×C₄

D₄×C₁₀₉

yes

—

155

C₂×M₁₆

—

156

C₂×M₁₆

—

157

C₄⋊Q₈

—

158

C₄⋊Q₈

—

159

C₄×D₄

—

160

C₄×D₄

—

161

C₂×C₄⋊C₄

—

162

C₂×C₄⋊C₄

—

163

C₂×C₄⋊C₄

—

164

C₂×C₄⋊C₄

D₄×C₁₀₉

yes

characteristic

165

C₂×C₂²⋊C₄

—

166

C₂×C₂²⋊C₄

C₂×C₄×C₁₀₉

yes

—

167

C₂×C₄²

—

168

C₂×C₄²

C₂³×C₁₀₉

yes

characteristic

169

C₄⋊C₈

—

170

C₄⋊C₈

—

171

C₄⋊C₈

D₄×C₁₀₉

yes

—

172

#100

—

173

C₂²⋊C₈

—

174

C₂²⋊C₈

—

175

C₄×C₈

—

176

#95

—

177

#95

C₂×C₄×C₁₀₉

yes

—

178

C₂⁴

C₂²⋊C₄×C₁₀₉

yes

—

179

C₂×Q₈

—

180

C₂×Q₈

—

181

C₂×D₄

—

182

C₂×D₄

—

183

C₂×D₄

—

184

C₂×D₄

—

185

C₂²×C₄

—

186

C₂²×C₄

—

187

C₂²×C₄

—

188

C₂²×C₄

—

189

C₂²×C₄

—

190

C₂²×C₄

C₂²⋊C₄×C₁₀₉

yes

—

191

C₂²×C₄

C₂²×C₄×C₁₀₉

yes

characteristic

192

C₂²×C₄

C₂×D₄×C₁₀₉

yes

characteristic

193

C₂²×C₄

C₂×D₄×C₁₀₉

yes

characteristic

194

M₁₆

—

195

M₁₆

—

196

M₁₆

—

197

C₂×C₈

—

198

C₂×C₈

—

199

C₄⋊C₄

—

200

C₄⋊C₄

—

201

C₄⋊C₄

—

202

C₄⋊C₄

—

203

C₄⋊C₄

—

204

C₂²⋊C₄

—

205

C₂²⋊C₄

—

206

C₄²

—

207

C₄²

—

208

C₄²

C₂×D₄×C₁₀₉

yes

—

209

C₄²

#41×C₁₀₉

yes

—

210

C₂³

—

211

C₂³

—

212

C₂³

—

213

C₂³

—

214

C₂³

—

215

C₂³

C₂×C₂²⋊C₄×C₁₀₉

yes

characteristic

216

Q₈

—

217

D₄

—

218

D₄

—

219

C₂×C₄

—

220

C₂×C₄

—

221

C₂×C₄

—

222

C₂×C₄

—

223

C₂×C₄

—

224

C₂×C₄

—

225

C₂×C₄

—

226

C₂×C₄

—

227

C₂×C₄

—

228

C₂×C₄

—

229

C₂×C₄

—

230

C₂×C₄

C₄×D₄×C₁₀₉

yes

—

231

C₂×C₄

#120×C₁₀₉

yes

characteristic

232

C₂×C₄

#121×C₁₀₉

yes

—

233

C₂×C₄

#123×C₁₀₉

yes

characteristic

234

C₂×C₄

AGL(1,ℤ/8)×C₁₀₉

yes

characteristic

235

C₂×C₄

#137×C₁₀₉

yes

—

236

C₂×C₄

#137×C₁₀₉

yes

characteristic

237

C₈

—

238

C₈

—

239

C₂²

—

240

C₂²

—

241

C₂²

—

242

C₂²

—

243

C₂²

—

244

C₂²

—

245

C₂²

—

246

C₂²

#386×C₁₀₉

yes

characteristic

247

C₂²

#417×C₁₀₉

yes

characteristic

248

C₂²

#419×C₁₀₉

yes

characteristic

249

C₄

—

250

C₄

—

251

C₄

—

252

C₄

—

253

C₄

—

254

C₄

#455×C₁₀₉

yes

—

255

C₄

#468×C₁₀₉

yes

—

256

C₄

#470×C₁₀₉

yes

—

257

C₂

—

258

C₂

—

259

C₂

—

260

C₂

C₁₀₉×#1885

yes

characteristic

261

C₂

C₁₀₉×#1894

yes

characteristic

262

C₂

C₁₀₉×#1914

yes

characteristic

	subgroup	G/H
center, socle	C₂²×C₁₀₉	#386
commutator	C₂²×C₄	C₂²×C₄×C₁₀₉
Frattini, cosocle	C₂×C₄²	C₂³×C₁₀₉
prime radical	C₂²×D₄×C₁₀₉	D₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₁₀₉ ◁ C₂×C₄²×C₁₀₉ ◁ G

subgroup

G/H

center, socle

C₂²×C₁₀₉

#386

commutator

C₂²×C₄

C₂²×C₄×C₁₀₉

Frattini, cosocle

C₂×C₄²

C₂³×C₁₀₉

prime radical

C₂²×D₄×C₁₀₉

D₄

LCS

G ▷ C₂²×C₄ ▷ C₂² ▷ C₁

UCS

C₁ ◁ C₂²×C₁₀₉ ◁ C₂×C₄²×C₁₀₉ ◁ G

N	H ≅ G/N	type
4. C₁₀₉×#1548	C₂	semidirect
5. C₁₀₉×#1303	C₂	semidirect
125. #9399	C₁₀₉	direct (N × H)
135. #1548	C₂×C₁₀₉	semidirect
136. #1303	C₂×C₁₀₉	semidirect
137. C₁₀₉	#9399	direct (N × H)

H ≅ G/N

type

4. C₁₀₉×#1548

C₂

semidirect

5. C₁₀₉×#1303

C₂

semidirect

125. #9399

C₁₀₉

direct (N × H)

135. #1548

C₂×C₁₀₉

semidirect

136. #1303

C₂×C₁₀₉

semidirect

137. C₁₀₉

#9399

direct (N × H)

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

subgroup	index
C₁₀₉×#2975	2
C₁₀₉×#2563	2
C₁₀₉×#1800	2
C₁₀₉×#1548	2
C₁₀₉×#1303	2
#9399	109

subgroup

index

C₁₀₉×#2975

C₁₀₉×#2563

C₁₀₉×#1800

C₁₀₉×#1548

C₁₀₉×#1303

#9399

109

Subgroups — C₁₀₉×#9399 #42127393

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C109×#9399 #42127393

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₀₉×#9399 #42127393