Group #42127395 — subgroups

Subgroups — C₁₀₉×#9401 #42127395

Order 27904.

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

normal subgroups only

This group has 534 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×#2547	C₂	1	1	yes	maximal, characteristic
3	C₁₀₉×#1800	C₂	1	1	yes	maximal, characteristic
4	C₁₀₉×#1546	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×#124×C₁₀₉	—	2	1	no	—
8	C₂×C₄×D₄×C₁₀₉	C₂²	1	1	yes	characteristic
9	#436×C₁₀₉	—	2	4	no	—
10	#431×C₁₀₉	C₂²	1	2	yes	—
11	#423×C₁₀₉	C₂²	1	2	yes	—
12	C₂×#100×C₁₀₉	—	2	1	no	—
13	#407×C₁₀₉	—	2	2	no	—
14	#397×C₁₀₉	C₄	1	4	yes	—
15	#391×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	#332×C₁₀₉	—	2	4	no	—
20	C₂²×Q₈×C₁₀₉	—	2	1	no	—
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	#124×C₁₀₉	—	4	4	no	—
28	C₄×D₄×C₁₀₉	—	2	2	no	—
29	C₄×D₄×C₁₀₉	—	2	2	no	—
30	C₂×C₄⋊C₄×C₁₀₉	—	2	1	no	—
31	C₂×C₄⋊C₄×C₁₀₉	—	2	1	no	—
32	C₂×C₄⋊C₄×C₁₀₉	C₂×C₄	1	2	yes	—
33	C₂×C₄⋊C₄×C₁₀₉	D₄	1	1	yes	characteristic
34	C₂×C₂²⋊C₄×C₁₀₉	—	2	4	no	—
35	C₂×C₂²⋊C₄×C₁₀₉	C₂×C₄	1	2	yes	—
36	C₂×C₄²×C₁₀₉	—	2	1	no	—
37	C₂×C₄²×C₁₀₉	C₂³	1	1	yes	characteristic
38	C₄⋊C₈×C₁₀₉	—	2	2	no	—
39	C₄⋊C₈×C₁₀₉	—	2	2	no	—
40	C₄⋊C₈×C₁₀₉	D₄	1	4	yes	—
41	#100×C₁₀₉	—	4	2	no	—
42	C₂²⋊C₈×C₁₀₉	—	2	4	no	—
43	C₂²⋊C₈×C₁₀₉	—	2	2	no	—
44	C₄×C₈×C₁₀₉	—	2	2	no	—
45	#95×C₁₀₉	—	2	4	no	—
46	#95×C₁₀₉	C₂×C₄	1	2	yes	—
47	C₂⁴×C₁₀₉	C₂²⋊C₄	1	2	yes	—
48	C₂×Q₈×C₁₀₉	—	4	2	no	—
49	C₂×Q₈×C₁₀₉	—	4	2	no	—
50	C₂×Q₈×C₁₀₉	—	4	1	no	—
51	C₂×Q₈×C₁₀₉	—	4	1	no	—
52	C₂×D₄×C₁₀₉	—	2	2	no	—
53	C₂×D₄×C₁₀₉	—	2	1	no	—
54	C₂×D₄×C₁₀₉	—	2	2	no	—
55	C₂×D₄×C₁₀₉	—	2	1	no	—
56	C₂²×C₄×C₁₀₉	—	2	2	no	—
57	C₂²×C₄×C₁₀₉	—	2	2	no	—
58	C₂²×C₄×C₁₀₉	—	2	2	no	—
59	C₂²×C₄×C₁₀₉	—	2	2	no	—
60	C₂²×C₄×C₁₀₉	—	4	4	no	—
61	C₂²×C₄×C₁₀₉	C₂²⋊C₄	1	2	yes	—
62	C₂²×C₄×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
63	C₂²×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
64	C₂²×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
65	M₁₆×C₁₀₉	—	4	2	no	—
66	M₁₆×C₁₀₉	—	4	2	no	—
67	M₁₆×C₁₀₉	—	8	4	no	—
68	C₂×C₈×C₁₀₉	—	2	4	no	—
69	C₂×C₈×C₁₀₉	—	4	2	no	—
70	C₄⋊C₄×C₁₀₉	—	2	2	no	—
71	C₄⋊C₄×C₁₀₉	—	2	4	no	—
72	C₄⋊C₄×C₁₀₉	—	4	2	no	—
73	C₄⋊C₄×C₁₀₉	—	4	1	no	—
74	C₄⋊C₄×C₁₀₉	—	4	1	no	—
75	C₂²⋊C₄×C₁₀₉	—	2	4	no	—
76	C₂²⋊C₄×C₁₀₉	—	4	8	no	—
77	C₄²×C₁₀₉	—	4	2	no	—
78	C₄²×C₁₀₉	C₂×D₄	1	2	yes	—
79	C₄²×C₁₀₉	#41	1	2	yes	—
80	C₂³×C₁₀₉	—	2	2	no	—
81	C₂³×C₁₀₉	—	2	2	no	—
82	C₂³×C₁₀₉	—	2	2	no	—
83	C₂³×C₁₀₉	—	4	2	no	—
84	C₂³×C₁₀₉	—	4	2	no	—
85	C₂³×C₁₀₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
86	Q₈×C₁₀₉	—	8	2	no	—
87	Q₈×C₁₀₉	—	8	2	no	—
88	D₄×C₁₀₉	—	4	2	no	—
89	D₄×C₁₀₉	—	4	2	no	—
90	C₂×C₄×C₁₀₉	—	2	2	no	—
91	C₂×C₄×C₁₀₉	—	2	2	no	—
92	C₂×C₄×C₁₀₉	—	4	1	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	—	4	2	no	—
98	C₂×C₄×C₁₀₉	—	4	2	no	—
99	C₂×C₄×C₁₀₉	—	4	2	no	—
100	C₂×C₄×C₁₀₉	—	4	2	no	—
101	C₂×C₄×C₁₀₉	—	8	2	no	—
102	C₂×C₄×C₁₀₉	C₄×D₄	1	2	yes	—
103	C₂×C₄×C₁₀₉	#120	1	1	yes	characteristic
104	C₂×C₄×C₁₀₉	#121	1	2	yes	—
105	C₂×C₄×C₁₀₉	#123	1	1	yes	characteristic
106	C₂×C₄×C₁₀₉	AGL(1,ℤ/8)	1	2	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	—	2	1	no	—
112	C₂²×C₁₀₉	—	2	1	no	—
113	C₂²×C₁₀₉	—	4	4	no	—
114	C₂²×C₁₀₉	—	4	2	no	—
115	C₂²×C₁₀₉	—	4	2	no	—
116	C₂²×C₁₀₉	—	4	2	no	—
117	C₂²×C₁₀₉	—	8	2	no	—
118	C₂²×C₁₀₉	#386	1	1	yes	characteristic
119	C₂²×C₁₀₉	#417	1	1	yes	characteristic
120	C₂²×C₁₀₉	#418	1	1	yes	characteristic
121	C₄×C₁₀₉	—	2	1	no	—
122	C₄×C₁₀₉	—	2	2	no	—
123	C₄×C₁₀₉	—	4	2	no	—
124	C₄×C₁₀₉	—	8	2	no	—
125	C₄×C₁₀₉	—	8	2	no	—
126	C₄×C₁₀₉	#455	1	2	yes	—
127	C₄×C₁₀₉	#468	1	2	yes	—
128	C₄×C₁₀₉	#469	1	2	yes	—
129	#9401	C₁₀₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
130	C₂×C₁₀₉	—	2	1	no	—
131	C₂×C₁₀₉	—	2	1	no	—
132	C₂×C₁₀₉	—	8	2	no	—
133	C₂×C₁₀₉	#1885	1	1	yes	characteristic
134	C₂×C₁₀₉	#1895	1	1	yes	characteristic
135	C₂×C₁₀₉	#1915	1	1	yes	characteristic
136	#2975	C₂×C₁₀₉	1	1	yes	characteristic
137	#2547	C₂×C₁₀₉	1	1	yes	characteristic
138	#1800	C₂×C₁₀₉	1	1	yes	characteristic
139	#1546	C₂×C₁₀₉	1	2	yes	complemented
140	#1303	C₂×C₁₀₉	1	2	yes	complemented
141	C₁₀₉	#9401	1	1	yes	characteristic, direct product factor, Sylow
142	C₂×C₄⋊Q₈	C₂²×C₁₀₉	1	1	yes	characteristic
143	C₂×#124	—	2	1	no	—
144	C₂×C₄×D₄	C₂²×C₁₀₉	1	1	yes	characteristic
145	#436	—	2	4	no	—
146	#431	C₂²×C₁₀₉	1	2	yes	—
147	#423	C₂²×C₁₀₉	1	2	yes	—
148	C₂×#100	—	2	1	no	—
149	#407	—	2	2	no	—
150	#397	C₄×C₁₀₉	1	4	yes	—
151	#391	—	2	1	no	—
152	#380	—	2	2	no	—
153	C₄×C₄⋊C₄	C₂²×C₁₀₉	1	1	yes	characteristic
154	#343	—	2	1	no	—
155	#332	—	2	4	no	—
156	C₂²×Q₈	—	2	1	no	—
157	C₂²×D₄	D₄×C₁₀₉	1	1	yes	characteristic
158	C₂³×C₄	D₄×C₁₀₉	1	2	yes	—
159	C₂×M₁₆	—	2	4	no	—
160	C₂×M₁₆	—	2	2	no	—
161	C₄⋊Q₈	—	2	2	no	—
162	C₄⋊Q₈	—	2	2	no	—
163	#124	—	4	4	no	—
164	C₄×D₄	—	2	2	no	—
165	C₄×D₄	—	2	2	no	—
166	C₂×C₄⋊C₄	—	2	1	no	—
167	C₂×C₄⋊C₄	—	2	1	no	—
168	C₂×C₄⋊C₄	C₂×C₄×C₁₀₉	1	2	yes	—
169	C₂×C₄⋊C₄	D₄×C₁₀₉	1	1	yes	characteristic
170	C₂×C₂²⋊C₄	—	2	4	no	—
171	C₂×C₂²⋊C₄	C₂×C₄×C₁₀₉	1	2	yes	—
172	C₂×C₄²	—	2	1	no	—
173	C₂×C₄²	C₂³×C₁₀₉	1	1	yes	characteristic
174	C₄⋊C₈	—	2	2	no	—
175	C₄⋊C₈	—	2	2	no	—
176	C₄⋊C₈	D₄×C₁₀₉	1	4	yes	—
177	#100	—	4	2	no	—
178	C₂²⋊C₈	—	2	4	no	—
179	C₂²⋊C₈	—	2	2	no	—
180	C₄×C₈	—	2	2	no	—
181	#95	—	2	4	no	—
182	#95	C₂×C₄×C₁₀₉	1	2	yes	—
183	C₂⁴	C₂²⋊C₄×C₁₀₉	1	2	yes	—
184	C₂×Q₈	—	4	2	no	—
185	C₂×Q₈	—	4	2	no	—
186	C₂×Q₈	—	4	1	no	—
187	C₂×Q₈	—	4	1	no	—
188	C₂×D₄	—	2	2	no	—
189	C₂×D₄	—	2	1	no	—
190	C₂×D₄	—	2	2	no	—
191	C₂×D₄	—	2	1	no	—
192	C₂²×C₄	—	2	2	no	—
193	C₂²×C₄	—	2	2	no	—
194	C₂²×C₄	—	2	2	no	—
195	C₂²×C₄	—	2	2	no	—
196	C₂²×C₄	—	4	4	no	—
197	C₂²×C₄	C₂²⋊C₄×C₁₀₉	1	2	yes	—
198	C₂²×C₄	C₂²×C₄×C₁₀₉	1	1	yes	characteristic
199	C₂²×C₄	C₂×D₄×C₁₀₉	1	1	yes	characteristic
200	C₂²×C₄	C₂×D₄×C₁₀₉	1	1	yes	characteristic
201	M₁₆	—	4	2	no	—
202	M₁₆	—	4	2	no	—
203	M₁₆	—	8	4	no	—
204	C₂×C₈	—	2	4	no	—
205	C₂×C₈	—	4	2	no	—
206	C₄⋊C₄	—	2	2	no	—
207	C₄⋊C₄	—	2	4	no	—
208	C₄⋊C₄	—	4	2	no	—
209	C₄⋊C₄	—	4	1	no	—
210	C₄⋊C₄	—	4	1	no	—
211	C₂²⋊C₄	—	2	4	no	—
212	C₂²⋊C₄	—	4	8	no	—
213	C₄²	—	4	2	no	—
214	C₄²	C₂×D₄×C₁₀₉	1	2	yes	—
215	C₄²	#41×C₁₀₉	1	2	yes	—
216	C₂³	—	2	2	no	—
217	C₂³	—	2	2	no	—
218	C₂³	—	2	2	no	—
219	C₂³	—	4	2	no	—
220	C₂³	—	4	2	no	—
221	C₂³	C₂×C₂²⋊C₄×C₁₀₉	1	1	yes	characteristic
222	Q₈	—	8	2	no	—
223	Q₈	—	8	2	no	—
224	D₄	—	4	2	no	—
225	D₄	—	4	2	no	—
226	C₂×C₄	—	2	2	no	—
227	C₂×C₄	—	2	2	no	—
228	C₂×C₄	—	4	1	no	—
229	C₂×C₄	—	4	4	no	—
230	C₂×C₄	—	4	2	no	—
231	C₂×C₄	—	4	2	no	—
232	C₂×C₄	—	4	2	no	—
233	C₂×C₄	—	4	2	no	—
234	C₂×C₄	—	4	2	no	—
235	C₂×C₄	—	4	2	no	—
236	C₂×C₄	—	4	2	no	—
237	C₂×C₄	—	8	2	no	—
238	C₂×C₄	C₄×D₄×C₁₀₉	1	2	yes	—
239	C₂×C₄	#120×C₁₀₉	1	1	yes	characteristic
240	C₂×C₄	#121×C₁₀₉	1	2	yes	—
241	C₂×C₄	#123×C₁₀₉	1	1	yes	characteristic
242	C₂×C₄	AGL(1,ℤ/8)×C₁₀₉	1	2	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	—	2	1	no	—
248	C₂²	—	2	1	no	—
249	C₂²	—	4	4	no	—
250	C₂²	—	4	2	no	—
251	C₂²	—	4	2	no	—
252	C₂²	—	4	2	no	—
253	C₂²	—	8	2	no	—
254	C₂²	#386×C₁₀₉	1	1	yes	characteristic
255	C₂²	#417×C₁₀₉	1	1	yes	characteristic
256	C₂²	#418×C₁₀₉	1	1	yes	characteristic
257	C₄	—	2	2	no	—
258	C₄	—	2	1	no	—
259	C₄	—	4	2	no	—
260	C₄	—	8	2	no	—
261	C₄	—	8	2	no	—
262	C₄	#455×C₁₀₉	1	2	yes	—
263	C₄	#468×C₁₀₉	1	2	yes	—
264	C₄	#469×C₁₀₉	1	2	yes	—
265	C₂	—	2	1	no	—
266	C₂	—	2	1	no	—
267	C₂	—	8	2	no	—
268	C₂	C₁₀₉×#1885	1	1	yes	characteristic
269	C₂	C₁₀₉×#1895	1	1	yes	characteristic
270	C₂	C₁₀₉×#1915	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₁₀₉×#2975

C₂

yes

maximal, characteristic

C₁₀₉×#2547

C₂

yes

maximal, characteristic

C₁₀₉×#1800

C₂

yes

maximal, characteristic

C₁₀₉×#1546

C₂

yes

maximal, complemented

C₁₀₉×#1303

C₂

yes

maximal, complemented

C₂×C₄⋊Q₈×C₁₀₉

C₂²

yes

characteristic

C₂×#124×C₁₀₉

—

C₂×C₄×D₄×C₁₀₉

C₂²

yes

characteristic

#436×C₁₀₉

—

#431×C₁₀₉

C₂²

yes

—

#423×C₁₀₉

C₂²

yes

—

C₂×#100×C₁₀₉

—

#407×C₁₀₉

—

#397×C₁₀₉

C₄

yes

—

#391×C₁₀₉

—

#380×C₁₀₉

—

C₄×C₄⋊C₄×C₁₀₉

C₂²

yes

characteristic

#343×C₁₀₉

—

#332×C₁₀₉

—

C₂²×Q₈×C₁₀₉

—

C₂²×D₄×C₁₀₉

D₄

yes

characteristic

C₂³×C₄×C₁₀₉

D₄

yes

—

C₂×M₁₆×C₁₀₉

—

C₂×M₁₆×C₁₀₉

—

C₄⋊Q₈×C₁₀₉

—

C₄⋊Q₈×C₁₀₉

—

#124×C₁₀₉

—

C₄×D₄×C₁₀₉

—

C₄×D₄×C₁₀₉

—

C₂×C₄⋊C₄×C₁₀₉

—

C₂×C₄⋊C₄×C₁₀₉

—

C₂×C₄⋊C₄×C₁₀₉

C₂×C₄

yes

—

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×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

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

—

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

—

100

C₂×C₄×C₁₀₉

—

101

C₂×C₄×C₁₀₉

—

102

C₂×C₄×C₁₀₉

C₄×D₄

yes

—

103

C₂×C₄×C₁₀₉

#120

yes

characteristic

104

C₂×C₄×C₁₀₉

#121

yes

—

105

C₂×C₄×C₁₀₉

#123

yes

characteristic

106

C₂×C₄×C₁₀₉

AGL(1,ℤ/8)

yes

—

107

C₂×C₄×C₁₀₉

AGL(1,ℤ/8)

yes

characteristic

108

C₂×C₄×C₁₀₉

#137

yes

characteristic

109

C₈×C₁₀₉

—

110

C₈×C₁₀₉

—

111

C₂²×C₁₀₉

—

112

C₂²×C₁₀₉

—

113

C₂²×C₁₀₉

—

114

C₂²×C₁₀₉

—

115

C₂²×C₁₀₉

—

116

C₂²×C₁₀₉

—

117

C₂²×C₁₀₉

—

118

C₂²×C₁₀₉

#386

yes

characteristic

119

C₂²×C₁₀₉

#417

yes

characteristic

120

C₂²×C₁₀₉

#418

yes

characteristic

121

C₄×C₁₀₉

—

122

C₄×C₁₀₉

—

123

C₄×C₁₀₉

—

124

C₄×C₁₀₉

—

125

C₄×C₁₀₉

—

126

C₄×C₁₀₉

#455

yes

—

127

C₄×C₁₀₉

#468

yes

—

128

C₄×C₁₀₉

#469

yes

—

129

#9401

C₁₀₉

yes

maximal, characteristic, direct product factor, Sylow

130

C₂×C₁₀₉

—

131

C₂×C₁₀₉

—

132

C₂×C₁₀₉

—

133

C₂×C₁₀₉

#1885

yes

characteristic

134

C₂×C₁₀₉

#1895

yes

characteristic

135

C₂×C₁₀₉

#1915

yes

characteristic

136

#2975

C₂×C₁₀₉

yes

characteristic

137

#2547

C₂×C₁₀₉

yes

characteristic

138

#1800

C₂×C₁₀₉

yes

characteristic

139

#1546

C₂×C₁₀₉

yes

complemented

140

#1303

C₂×C₁₀₉

yes

complemented

141

C₁₀₉

#9401

yes

characteristic, direct product factor, Sylow

142

C₂×C₄⋊Q₈

C₂²×C₁₀₉

yes

characteristic

143

C₂×#124

—

144

C₂×C₄×D₄

C₂²×C₁₀₉

yes

characteristic

145

#436

—

146

#431

C₂²×C₁₀₉

yes

—

147

#423

C₂²×C₁₀₉

yes

—

148

C₂×#100

—

149

#407

—

150

#397

C₄×C₁₀₉

yes

—

151

#391

—

152

#380

—

153

C₄×C₄⋊C₄

C₂²×C₁₀₉

yes

characteristic

154

#343

—

155

#332

—

156

C₂²×Q₈

—

157

C₂²×D₄

D₄×C₁₀₉

yes

characteristic

158

C₂³×C₄

D₄×C₁₀₉

yes

—

159

C₂×M₁₆

—

160

C₂×M₁₆

—

161

C₄⋊Q₈

—

162

C₄⋊Q₈

—

163

#124

—

164

C₄×D₄

—

165

C₄×D₄

—

166

C₂×C₄⋊C₄

—

167

C₂×C₄⋊C₄

—

168

C₂×C₄⋊C₄

C₂×C₄×C₁₀₉

yes

—

169

C₂×C₄⋊C₄

D₄×C₁₀₉

yes

characteristic

170

C₂×C₂²⋊C₄

—

171

C₂×C₂²⋊C₄

C₂×C₄×C₁₀₉

yes

—

172

C₂×C₄²

—

173

C₂×C₄²

C₂³×C₁₀₉

yes

characteristic

174

C₄⋊C₈

—

175

C₄⋊C₈

—

176

C₄⋊C₈

D₄×C₁₀₉

yes

—

177

#100

—

178

C₂²⋊C₈

—

179

C₂²⋊C₈

—

180

C₄×C₈

—

181

#95

—

182

#95

C₂×C₄×C₁₀₉

yes

—

183

C₂⁴

C₂²⋊C₄×C₁₀₉

yes

—

184

C₂×Q₈

—

185

C₂×Q₈

—

186

C₂×Q₈

—

187

C₂×Q₈

—

188

C₂×D₄

—

189

C₂×D₄

—

190

C₂×D₄

—

191

C₂×D₄

—

192

C₂²×C₄

—

193

C₂²×C₄

—

194

C₂²×C₄

—

195

C₂²×C₄

—

196

C₂²×C₄

—

197

C₂²×C₄

C₂²⋊C₄×C₁₀₉

yes

—

198

C₂²×C₄

C₂²×C₄×C₁₀₉

yes

characteristic

199

C₂²×C₄

C₂×D₄×C₁₀₉

yes

characteristic

200

C₂²×C₄

C₂×D₄×C₁₀₉

yes

characteristic

201

M₁₆

—

202

M₁₆

—

203

M₁₆

—

204

C₂×C₈

—

205

C₂×C₈

—

206

C₄⋊C₄

—

207

C₄⋊C₄

—

208

C₄⋊C₄

—

209

C₄⋊C₄

—

210

C₄⋊C₄

—

211

C₂²⋊C₄

—

212

C₂²⋊C₄

—

213

C₄²

—

214

C₄²

C₂×D₄×C₁₀₉

yes

—

215

C₄²

#41×C₁₀₉

yes

—

216

C₂³

—

217

C₂³

—

218

C₂³

—

219

C₂³

—

220

C₂³

—

221

C₂³

C₂×C₂²⋊C₄×C₁₀₉

yes

characteristic

222

Q₈

—

223

Q₈

—

224

D₄

—

225

D₄

—

226

C₂×C₄

—

227

C₂×C₄

—

228

C₂×C₄

—

229

C₂×C₄

—

230

C₂×C₄

—

231

C₂×C₄

—

232

C₂×C₄

—

233

C₂×C₄

—

234

C₂×C₄

—

235

C₂×C₄

—

236

C₂×C₄

—

237

C₂×C₄

—

238

C₂×C₄

C₄×D₄×C₁₀₉

yes

—

239

C₂×C₄

#120×C₁₀₉

yes

characteristic

240

C₂×C₄

#121×C₁₀₉

yes

—

241

C₂×C₄

#123×C₁₀₉

yes

characteristic

242

C₂×C₄

AGL(1,ℤ/8)×C₁₀₉

yes

—

243

C₂×C₄

AGL(1,ℤ/8)×C₁₀₉

yes

characteristic

244

C₂×C₄

#137×C₁₀₉

yes

characteristic

245

C₈

—

246

C₈

—

247

C₂²

—

248

C₂²

—

249

C₂²

—

250

C₂²

—

251

C₂²

—

252

C₂²

—

253

C₂²

—

254

C₂²

#386×C₁₀₉

yes

characteristic

255

C₂²

#417×C₁₀₉

yes

characteristic

256

C₂²

#418×C₁₀₉

yes

characteristic

257

C₄

—

258

C₄

—

259

C₄

—

260

C₄

—

261

C₄

—

262

C₄

#455×C₁₀₉

yes

—

263

C₄

#468×C₁₀₉

yes

—

264

C₄

#469×C₁₀₉

yes

—

265

C₂

—

266

C₂

—

267

C₂

—

268

C₂

C₁₀₉×#1885

yes

characteristic

269

C₂

C₁₀₉×#1895

yes

characteristic

270

C₂

C₁₀₉×#1915

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×#1546	C₂	semidirect
5. C₁₀₉×#1303	C₂	semidirect
129. #9401	C₁₀₉	direct (N × H)
139. #1546	C₂×C₁₀₉	semidirect
140. #1303	C₂×C₁₀₉	semidirect
141. C₁₀₉	#9401	direct (N × H)

H ≅ G/N

type

4. C₁₀₉×#1546

C₂

semidirect

5. C₁₀₉×#1303

C₂

semidirect

129. #9401

C₁₀₉

direct (N × H)

139. #1546

C₂×C₁₀₉

semidirect

140. #1303

C₂×C₁₀₉

semidirect

141. C₁₀₉

#9401

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×#2547	2
C₁₀₉×#1800	2
C₁₀₉×#1546	2
C₁₀₉×#1303	2
#9401	109

subgroup

index

C₁₀₉×#2975

C₁₀₉×#2547

C₁₀₉×#1800

C₁₀₉×#1546

C₁₀₉×#1303

#9401

109

Subgroups — C₁₀₉×#9401 #42127395

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C109×#9401 #42127395

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₀₉×#9401 #42127395